Flat mana isn't as useful as often as mana regeneration; lets compare it as health for a moment.
There's chogath & sion that rely on large flat health, then there's garen & mundo who rely on health regeneration. Sure chogath & sion can survive more of a beating, but if they aren't killed then they have to back and heal, loosing farm or map presence for teamfights. While garen & mundo can just back off for a moment, returning to full strength while remaining in view for pressure and xp. Who do you see more often in any given day? Garen.
Now back to mana; there's ryze & singed who want large mana pools, then there's most other mages that rely on spam to get anything done such as syndra & brand. Ryze has low mana costs and singed gains health based on maximum mana while having his ult restore mana, so both don't require mana regeneration to function in long term fights. However syndra & brand burn 1/4~1/3 of their mana pool in a single combo, if it misses or fails (which usually does against any hyper-mobile champ), then they're less of a threat as after a couple attempts they won't have power to fight back at all, and with base regeneration being so low, they'll have to return to base to get a significant amount back, again loosing map pressure and xp. Have you tried fighting from levels behind? It's painful at best, worse if it's against someone abusable like zed or fizz.
The tear items at most reduce costs by 25%, and don't stack, which doesn't help most champions that have triple digit mana costs that much, as base mana pools are only a few hundred, which ends up just being an extra rotation of spells until it's fully charged. The lost chapter (and items it's builds into) are useless late game, as you won't be leveling up anymore; it's a 'win more' item, only helping if you're already crushing your lane opponent; if you're being crushed it won't help as you're often having to back or die.
I already said the grail was an exception, but it got both it's ability power & magic resist lowered without a price reduction; a nerf and riot claimed it to be only a support item. But I agree with you that it should be an acceptable mage item instead of relegated to supports only, I'd still use it on any defensive mage (such as swain or singed).
Then there's the rod of ages, nerfed repeatedly over the past year, and now it has 'eternity' which generates 15% of damage taken as mana; never mind it gives less flat health & flat mana than before. But wait they removed will of the ancients, leaving gunblade as the only magic spell-vamp item left, which only a few hybrids can make use of. There's no other sustainability left for mages that isn't suicidal, though swain & karthus work best in the enemy team like a melee bruiser, the rod of ages is pointless on ziggs and syndra as they won't have the survivability to gain any significant mana back without dying.
Oh but there's cooldown reduction everywhere! Useless when you don't have the mana to cast more than a single spell. Why do you think the doran's ring has become immensely popular again? It gives mana regeneration in 2 forms and for cheap!
I prefer looking at the in game statistics in their item sets, as that's more accurate on the days they don't update the wiki. Sure giant's belt isn't complete, but doesn't negate the fact the eye items have terrible stats. I'm well aware it's to prevent other classes from abusing them, but when supports won't even build them (eye of oasis, ohmwrecker, zeke's herald, ext) something is wrong.
Bloodthirster used to be a core marksmen item back when its lifesteal was a stacking effect (gained lifesteal % each auto attack), still see it & similar used in most auto-attack reliant champions such as vayne with a ruined king's blade. Assassins are supposed to counter marksmen, the champions that used to be the only reliable class to tear down towers; not mages or tanks. They shouldn't need any armor penetration since marksmen don't build armor. Sterak's wasn't nerfed logically; they gutted it because of marksmen from taking it, but should've nerfed it so that only tanks would benefit from it; it's next to unused outside of a few problem cases. Mages can only take towers quickly because they have some of the highest waveclear, and that towers are still pathetic. Unless drastically ahead, the mage won't be as much of a threat as the marksmen. Assassins are played mid because the minions arrive faster than top or bot, thus leveling up & getting gold faster, while having relative access to top, bottom, and jungle.
The gold difference between supports & their carry used to be around only a grand; now it's 4 thousand, that's nearly 2 items behind! I'm not saying fixing the assist system will help only supports, I'm saying supports are behind the most and that the assist system is only part of it. The majority is because the gold generators are easily countered by both the enemy team & your own team, because the items give gold situationaly.
Indeed but most any class is double screwed because riot has forced the game to speed up. They turned snowballing into an avalanche solely for the sake of 'pro' games, which have ironically become even more boring than before; 10 minutes of farming, one kill, then a teamfight followed by so many objectives that the other team can't catch up, then the game is over.