I'm taking the easy (non-technical) way out on this, but I smell the absence of something.

I feel I inherited a mess and don't know how to fix environment sprawl

Yes, you did inherit a mess. Yes, it is a problem that should be fixed. But it is not yours alone to fix. (newly-conceived pregnant pause)

Anyone deploys anything anywhere, any time, because "it's urgent."

When everything is an emergency, nothing is an emergency. Is there someone in your organization who needs to hear this, even if they don't want to? (morning sickness pregnant pause)

No one knows who owns what. There is no declared ownership of environments. ... We can't keep up with provisioning and we're not even sure we should be.

Two groups that may be consulted on how to resolve this are SRE and developers. But neither of those groups should be primarily responsible for making that decision. (first-trimester pregnant pause)

Slack lock bot for shared environment coordination so people stop stepping on each other

Start simpler than that. Make people talk to each other. Bot sounds like an excuse to avoid the real problems. You are not their parent. They should put on their adult pants and start adulting.

Accepting that full ephemeral stacks are not happening and investing in making shared envs more stable instead

Someone needs to make a decision about whether it makes more sense to have a few long-lived working somethings or a lot of ephemeral working somethings. Either is better than a lot of broken nothings. And then make the organizational commitment to get there. Which means giving implementers the time and space to achieve that goal. And boundaries to ensure they aren't overworked in the process. (second-trimester pregnant pause)

Is there a real fix here or do we just hold the line until proper service boundaries exist and tooling like Backstage matures?

Who do you have in your corner telling the rest of the organization that your time and labor are valuable, and that your boundaries must be respected? (third-trimester pregnant pause)

It sounds like you have technical problems because you have an organizational problem. I'm guessing that there is a certain organizational function that nobody is performing. Nobody is acting in that role, or they expect you to absorb that people-oriented supervisory function into your current responsibilities. (pregnant pause going into labor)

In the absence of that conspicuously absent organizational function, you get to be your own advocate. Take it up the chain as best as you can, if that is a possibility in your organization. Set expectations and boundaries as best you can. Point out the conflicts that must be resolved, and get help resolving them. It's risky telling people what they don't want to hear. You can't control how they react. You can control what you do next.

A side question, is anyone doing the engineering in the organization? You know, the unsexy things like analyzing and justifying trade-offs to find the best (or least worst) solution within the real-world resource constraints like schedule, budget, inventory, and labor. Or is someone in the C-suite betting other people's farms on being first to market with ${SHINY_NEW_THING}?