To your first bullet point, your own example of StarCraft. Rush strategies are usually so all-in that they win or lose in a couple of minutes. If they're successfully defended, the defender now has such an advantage that the rusher can't come back from it.

I actually don't know of a game that's ruined by an "aggression meta". I don't think I agree that it's a problem. Neither rushing nor turtling is incentivized in StarCraft. The push and pull that the designers wanted from a given match is the optimal way to play, and you'll find more success chasing that than either turtling or rushing.

I'm heavily invested in the fighting game scene, and the genre's been getting more and more "aggression mechanics" for a long time now; some might call them "neutral skips", skipping the part of the game where the two players try to approach each other. There's a clear reason for why they do this: it's way more fun to watch. Street Fighter V often devolved into two players left on their last pixel of health, since you can't kill with chip damage (for the most part), so it was a boring situation of both players fishing for a last hit as the clock ticks down. Now, whether it's Street Fighter, Mortal Kombat, or Guilty Gear, you have a meter that you use on offense and defense. Being offensive rewards you with more and allows you to be more offensive, and being defensive will drain it. You can still have that moment from SFV that was supposed to be tense, but now it's actually tense, because while that player is defending, the resource that prevents a checkmate situation is draining down, and when it's empty, it's basically game over.

Dark Souls, and other from soft souls likes (except Sekiro and Bloodborne).

You are encouraged to play cautiously and intently, otherwise you'll get slapped by concealed enemies, mobs with unexpected movesets, and being over aggressive and "greedy" during boss fights will end many an attempt. I love these games for that.

I haven't played Overwatch for a while but for a time there was a notorious meta called GOATS (3 tanks, 3 supports). It was an insanely aggressive meta that focused on rushing straight into the enemy team, tanking them, and killing them before they can react. The only way you can counter it is by also running the same team comp and hoping to kill them faster.

It ruined ranked games for a few months and the devs apparently had no idea how to fix it without nerfing tanks or supports hard - which would make playing them feel terrible. That's why OW added a role queue and enforced 2 damage, 2 tank, 2 support teams.

That said, I think aggressive metas are way better than turtling ones. Nobody wants to idle around and take pot shots until someone gets bored.

Some games that come to mind:

Dead by Daylight has an issue with killers that keep their focus on one of the four survivors, ignoring the core objectives and other players. Worse, it often works well. There are many videos out there of experienced teams that find karmic counters for this practice, helping the victim escape the killer to some completely unknown location on the map, and often leaving the killer late-game with little to work with.

Texas Chainsaw Massacre (a 4v3 horror game), on the other hand, developed some issues where the prevailing strategies for the victims involve stacking up abilities that let them ignore attacks so there's no need to hide or move slowly. It ends up taking long enough for the family members to even strike them down that some will brute-force objectives right in the family's face. Part of the game's issues is, the maps are developed to be relatively tight, so there's fewer places for family to check, but it also made stealth strategies relatively ineffective.

An old favorite of mine for countering "Rush Meta" is in Team Fortress 2. For single players hoping to run past players to objectives, the Engineer's sentry locks on to them pretty quickly, and no matter how fast they're moving, it spells death within a certain bubble. Being automated, it also means no one has to camp for this to stay around. The sentries still die to inexperienced players that are making a unified push.

TF2's other "rush punisher" is the Heavy - a class with a low skill cap, but a high health pool. He deals ludicrous damage up close, but can't move quickly. So, he's most lethal to people that are running at/past him instead of attacking from a distance. He says it right in his intro - he can't outsmart people. He's just a strong presence in a push for anyone that doesn't have a plan to slow themselves down in order to deal the ton of damage needed to kill him. For a long time, in matches where the enemy team stuck to having 3 pyros rushing the frontline, my sole strategy was to pile up on Heavy, forcing the enemy team to consider ranged attackers like Demoman and Sniper, slowing the game down as a result.

Competitive Pokémon tends to go back and forth between times of "stall" (turtling) and hyper-offense (aggression) dominating the metagame, depending on which strategies and team builds players will find. Whenever one becomes dominant, fans of the other will constantly hound tournament runners to change tiering or ban certain pokemon to change it.

As for a "fun" game to go hyper-aggressive with zero HP, max damage, I've seen some YouTubers attempt "Danger Mario" runs in the first two Paper Mario games, maximizing FP and BP and never taking HP when leveling, keeping Mario in the "danger" zone where lots of evasion or damage badges will stay activated. They then rely on those badges, items and partner abilities to avoid taking damage.

Aggression should be part of a game, but shouldn't be the only way to play it. Obviously, when a game is optimized, it may be the best way to play (Monster Hunter and HAME speedruns come to mind), but a lot of great games try to design so that different archetypes can coexist and play off one another.

Street Fighter 6 encourages aggression. The Drive Meter system makes it so that turtling and blocking forever will end with you in blowout, taking chip damage and having worse frame disadvantage, as well as removing your ability to use Drive moves and opening you up for stuns. However, also hidden within the Drive System are some of the tools to deter mindless aggression. Drive Impacts are big moves with armor that lead into a full combo, so if you can read a braindead attack sequence, you can Drive Impact to absorb a hit, smack them, and then combo them for 35% of their life total. There are also parries, which can refill your drive meter.

Magic: The Gathering has tried to balance the various archetypes (Aggro, Midrange, Control, and Combo) so that every format should have at least 1 competitively viable deck in each meta archetype. Typically, Aggro will be too fast for a Control deck to stabilize and kill them before they can get their engine set up. But Midrange will trade just efficiently enough (with good 2-for-1 removal or creatures) to stop the aggression, and then start plopping out creatures that Aggro will have difficulty overcoming. And Combo often has nothing to fear from Aggro, since Aggro oftentimes can't interact with the game-winning combo pieces. And because of this system, Aggro decks have to have sideboard plans ready for whatever meta they expect at an event or tournament. Removal or protection to get over or under Midrange, and faster speed or other types of interaction to take down or disrupt Combo. Magic's systems (Mana/lands, instant speed removal, and even the variance that comes from being a card game) don't punish aggro directly, but they make sure that there are usually answers out there.