Demolish is getting removed right?

You're completely ignoring Demolish now, but okay.

It's actually pretty difficult to make use of Overheal on a marksman early in the game without a champion that heals for tons in a single go, like Bard or Soraka. It's nearly impossible in a lane with lots of poke or aggression, like Jhin, MF, Morgana, Bard, etc.. No Marksman builds Bloodthirster early (or should, outside of Draven) so that's a good portion of the game it isn't being used in that way. This does introduce the possibility of rushing an early bloodthirster on champs that normally wouldn't take it at all, but remember: Bloodthirster's stats don't actually offer much to almost any marksman's kit early on. It's much better to start stacking crit items beforehand, so that bloodthirster has more value when you get it. So let's say somebody rushes bloodthirster to get easy overshields for the shield bash proc. Since they don't have nearly as much attack speed, there's no way they'll be able to heal up to the overshield in the midst of a fight to get the constant procs. In fact, their lack of dps because of bloodthirster only offering its flat AD to damage, shield bash would then be trying to make up for the difference in dps, which is very difficult to do since it's not a giant on-hit effect, and it doesn't hit every auto attack, just every auto after your shield is gone and you get a new one. So if somebody doesn't manage to penetrate your overshield, you're not getting new procs unless you let it fade (which takes longer than it does to lose cs when you have lifesteal).

Also, it scales with only your health, so the damage proc won't even be that much. trying to build around Shield Bash on a marksman is just inefficient because there are no marksman items that give you bonus health early on aside from black cleaver and frozen mallet, and the health is the only thing in black cleaver giving you the synergy with the focus on shield bash, so... that's inefficient too. You might argue that the armor shred helps with lifestealing, but now you need black cleaver, and lifesteal, forcing you into blade of the ruined king to get the attack speed marksman MUST have.

Worst case scenario for marksman, you'll see tons of wonky builds that work weirdly well, but building around shield bash will require sacrificing so much dps that they won't be playing like a real marksman until very, very late in the game, or getting no benefit from shield bash until very, very late in the game at which point the miniscule damage from its scaling will make it negligible anyways. The only champ I see this maybe being a problem for is Draven because of the value he can pull from overheal and bloodthirster early on, and his ability to manipulate the timing of his receiving damage compared to when he dishes it out. But you know, everyone knew Kleptomancy was going to be absolutely ridiculous on a bunch of champions, but nerfing them in the proper ways brought them in line without having to do much about balancing klepto. It'll probably be the same deal with this, and it'll almost certainly be the same deal with the new Dark Harvest.

Just build the proper anti-tank items and group. A full bruiser team might win fights when both sides start at full health, but that's why champions like Brand, Xerath, Caitlyn, Veigar, and Ashe exist. So that the opposing side doesn't get to engage with their whole team at full health. Also build the appropriate anti-tank items. You're not trying to compete with huge burst, just dps and durability. Grievous wounds for regenerators, Liandry's and BotRK for huge beefy dudes, Void Staff/Last Whisper. One thing I've done that works hilariously well is to do bruiser builds on non-bruisers with some penetration, rather than raw damage with penetration. Once, when faced against a full tank team, I actually did a tanky Kindred build (largely because I had Grasp by accident) and it was mindblowingly effective. Other champs that would just be really, really good for an anti-tank comp are Graves, Vayne, Malzahar, Kog'Maw, and twitch, mages with big aoe cc like Veigar, Brand (the fear of a team of tanks having to deal with an endless brand ult is both hilarious and effective enough to count), Orianna, Bard, and of course, your own tanks/bruisers that are good against tanks, like Maokai, Darius, Gnar, Vi, Fiora, and Urgot. Don't make your goal just to try to dish out as much damage as you can, you also want to be able to survive them. Tanks will win longer fights because their durability allows them to get multiple ability rotations. You just need to be able to live long enough to get the same number of attacks/ability rotations, and your kit's built in dps should do the rest.

Or you can just play champs like jinx, Quinn, and Tryndamere that can fly around the map and push lanes/force objectives faster than a team of tanks can respond to, and deny them the 5v5 fights they want.

So, what you're saying is that assassins should be able to deal with tanks, and that ad carries are overtuned in what you're saying is already a tank dominant meta? Akali only seems busted because most people don't know how to play around her shroud cooldown and her energy management. Once you figure out timing and a little bit of wave management, it isn't hard to scare her long enough for you to outscale. Sion is agreeably overpowered right now because there's very little difference between a fed sion and a feeding sion. Most DPS tanks are in line with this, like Volibear, Darius, Urgot, Renekton, and other tanks that both have good early farming and high scaling base damage on abilities. But other ones like Nautilus Malphite, the ones that sacrifice damage for extra tankiness and utility, really need something like shield bash so that in a fight they actually have some oomph behind them.