toucanbuzz
No rule is inviolate
...BTW, is it intentional that staggered really hurts directly offensive characters, but doesn't impact buffers and healers? I'm actually good with it - buffers and healers give others a chance to shine so I support that type of play. But if you want it to be more even and affect them as well, at the very least have concentration break when you first gain staggered.
Yes, this isn't an ideal time to go offensive, and for game balance, it can't because we're allowing people to act at a time they normally are on the floor bleeding out and unconscious. We're handing out a huge boon that opens up a ton of options that weren't previously available. When you're unconscious, you have no options, and your allies have one option: heal your butt or you might die. But if you elect to be Staggered, you can Dodge, Disengage, use an item, whatever, and your allies might do other useful stuff depending on how the battle is going.
It's designed to give players more strategic power, BUT we also have to ensure we avoid abuse. Without exhaustion, and because we don't (and shouldn't) use spillover damage, there's some severe abuse that could go on. For example, an elephant gores Fighter for 20 damage, dropping him to 0hp. Fighter keeps going, Staggered, drinks a potion to heal 5 damage. Elephant gores Fighter for 20 damage, dropping him again to 0. Fighter keeps going on and drinks another potion. In short, without a penalty, Fighter could keep soaking an Elephant's 20-damage attack each round for the mere cost of a 5 point heal. Even if we impose other penalties like a cumulative -1 to everything, Fighter can use this system against certain foes to be a damage sponge.
As above, the power needs to be in the players' hands. Like the Diehard feat from 3rd edition, they choose whether to draw on reserves or slip into unconsciousness. If the idea of that level of Exhaustion is too much to bear, then play it like normal. You're knocked out. But if you're willing to accept some lasting repercussions, then stay in the fight a bit longer and do whatever it is you need to do.
Regarding Concentration, it would be in line that all Concentration saves be made when entering the condition and while in the condition be made with disadvantage. You may want to keep afloat to try and keep that Haste spell up on your allies, but the pain of actual injury makes it extremely difficult.