So, first off, there wasn't always "just one right" page in the old runes system. Perhaps certain champions in certain roles had a specific rune page that was right for most situations... But for the most part, you would switch rune pages. Sure, people could get by with a general rune page, and even I did that... But, there are a lot of knowledge in the old rune system. From switching just a couple runes for a specific matchup to change a specific situation that you'd only know and understand after playing that matchup dozens of times... to just preparing for a different situation defined by the entire game or champ choices.
I ran 12 different pages for jungling. I'd use a general page for mages, enchanter supports, tank supports, ADCs, and whatnot, but since I mained JG, I would run enough pages to cover every situation. Most people weren't aware of how best to use runes. Even had a buddy who thought AD runes were best for any champ that auto attacks. I was able to convince him that for a champion like Warwick, you'd rather have AS runes. (Old WW) Riot seemed to think that runes were "solved", and maybe they were, but there was an enormous amount of knowledge based on the old rune system. It wasn't as cut and dry as most people seem to think.
Riot had been looking to removed runes for a long time, though, and they finally did. Originally, this new "runes reforged" idea completely cut off the old rune system and just reworked Masteries, like they do every other year on average. We call them runes, but these are just masteries. After they realized that general stats attached to specific mastery trees aren't gonna work for every champion... they backed off on that decision and added the old runes back, but in a much, much more restricted way. We don't have Quints anymore, instead of being able to stack one type of stat (defense or damage), but less efficiently in certain slots. (Armor glyphs are weaker than armor seals) But, we still have a defense slot, an offense slot, and what they call a "flex" slot, that can sorta be one or the other. It's important that they allow us to pick the stats we need, because every champion is different and needs a unique combination. Now, many champions will end up with very similar runes, but that's okay.
When it comes to giving every champion a stat buff after they removed the old rune system... Well, the new runes give less stats than before when it comes to armor/damage/attack speed etc... And the entire game had been balanced around those stats we got from the old runes. So, they could have just made the jungle camps easier to kill in general, or give those stats to the champions that struggled. (Not every champ got the stats they needed, so it wasn't a net buff for everyone. Even if a champ got some stats, they might not have been enough stats to make up for it, while another champ did get the right stats, meaning the first champ is nerfed relative to the rest of the roster). It's possible if they changed the game instead of the champions, that you'd see picks that weren't strong enough before, getting picked in weird positions. So it's a much safer option to just give those stats back to the champions and try to start from where they were before, rather than creating a completely new environment to understand and balance. (Like awkward jungle picks because now a champ that couldn't clear, can, because the JG is easier).
By constantly switching the game, they promote adaptability. Adaptability is one of the skills that Riot expects you to have in order to climb. (Which is why an RNG element like elemental drakes was added to the game. It's not so RNG where a game is decided by that alone, because you're not going to lose a game cause the enemy team got all the Ocean drakes they need for their poke comp... you lost that game cause obviously you didn't have very much control at any stage of the game... Also, while ocean drakes promote poke comps, they also counter poke comps. That's pretty amazing).
The new runes are actually really interesting. In the past, Runes were mostly passive and uninteresting stats, and yeah, some of your arguments might hold up against those. (Old runes and masteries). However, one of the goals of the new runes reforged system is to give us something more than just passive stats. These are modifiers that we can interact with in game. They do promote different playstyles, and allow different playstyles. Sure, again, some champs might always go with the same type of runes, but, other champs might have half a dozen possible keystones that are effective in different situations, and among each of those keystones, you'll have different sub-runes for different situations. If I don't need to take Legend: Tenacity, I'm gonna take Legend: Alacrity if I need the dueling power, or Legend: Bloodlines if I need more sustain in fights. (I prefer sustain vs assassins, so I win if I keep fighting, an dueling power vs other fighters, so I get more auto's in).
When it comes to removing everything, you could say that about the entire game. Riot has said before that they could literally just adjust the HP of every champion to make the game 100% balanced. This is a game where all variables are under Riot's control, and so you can solve the balance "problem" with mathematics. But Riot has said previously that their goal isn't to balance this game. Their goal is to achieve perfect imbalance. Imbalance is interesting. It allows people to discover new things. If they wanted to make the game easier to balance, they could remove items, runes, and elemental dragons (go back to a regular dragon that just gives flat gold and nothing else, so you don't need to think about what those stats can do to every champion). But that's certainly not interesting.
Runes are unique, they allow us to express our skill and knowledge before the game even starts. And what's great about them, is that they're friendlier to new players. The other way to express our skill during champ select is in champ selection, team composition (which can require teamwork with a bunch of random people), or summoner spell selection. You don't even know all of the variables sometimes before you get to pick your champion, especially if you're first pick. If I first pick top, I pick my top laner who does well into most every matchup my other top laners will struggle with. Or I first pick my jungle champ that I can play well into anything. If I'm last pick, I can pick to support my comp and still not get countered. However, these are concepts that newer players will struggle to grasp for a long, long time. It takes a lot of experience and knowledge, and knowledge of just about every champion in the game, in order to understand this part of the game. Runes, on the other hand, are more specific to just your own champion, than the rest of the game. Sure, I'll pick a different rune depending on compositions, but I won't be punished very much if I pick Tenacity against a team with no Hard CC, and only silences and slows. It's still okay. Knowledge related to runes and which runes to pick is much easier to come by and find online, than knowledge on how to build a team composition or knowledge on specific lane matchups. It's a way to reward newer players for learning the game in a meaningful way.
Yeah, sometimes there are nightmare situations with runes when they make a change with unaccounted consequences. But the same can be said of any balance change/decision Riot could possibly make. They could leave the game as it is, and never change anything. What's cool about that, is that things we think are strong now, might not be strong a year from now, even if they never change the game itself. You can see this in a lot of older competitive games that no longer get patches. Tier lists change over the years because people discover new things. What might seem strong today, could be really weak once people understand how to play against it. But this is okay. Part of the skill of a game is having that knowledge and knowing what's the wrong decision and what's the right decision. If you could make any decision at all and it'll be okay, then that's not a very interesting or competitive situation. A lot of games actually purposefully have weak options to give newer players the opportunity to learn the difference between strong and weak. It's also rewarding when they first understand why you shouldn't pick that option. But Riot has opted to change their game every couple of weeks, and that's okay too.
To be fair to Riot, they actually do a great job balancing the game. I know this is an unpopular opinion, but it's the truth. When most champions and items, most of the time, have around a 50% winrate, or between 48% and 52%, they've done a pretty good job. Winrates aren't everything, though.
This game is incredibly complex, and to keep things within such a tight margin is difficult. The people constantly complaining about game balance probably really don't understand the first thing about balancing a game this complex. Yeah, changing something that seems strong might seem logical to most people, but there are so many other factors to consider, so many other things that change. Nerf this champ, and who else gets stronger as a result? And is that a healthy trade-off? Or should we buff something else that directly or indirectly counters that champ in some way, or buff something that makes that champ less of a great pick? There's a lot going on, so Riot takes their time to make sure they do it right. They don't always, though, but the people working at Riot are only human, and you can't expect them never to make a mistake.