Two things put me off, I do wonder if anyone else here has attempted to do it as a consultant?
Firstly, while I could quickly identify the cause of the slowdowns, the previous times I've done this I already had an intimate knowledge of the codebase so could confidently fix the problem. Without an intimate knowledge of the codebase, I fear I'd either muck up the fix, or that I'd have to rely on internal developers who, as you say, would deprioritize it or feel resentful at an outsider coming in and from their perspective effectively telling them their code is "bad" (even though I understand how these speed issues develop organically). Or would it be enough to just offer the diagnosis, but not the fix, and thus give ammo to the PMs/senior management to take the problem to their own developers and say "look, we've been told there's a problem and it's fixable, go fix it".
Secondly, how to handle getting access to their codebase/a proper debug environment that reflects production for such a short time. Again I fear it would turn into a "not my problem" scenario of trying to get a live environment setup from people who don't want to give it to you. Fixing these problems usually turns out to be fairly quick once you've identified the real cause. I can do a quick analysis just with access to a website (i.e. is it a back-end or front-end problem), but until you can run diagnostics you can't say exactly which loop/sql statement/js compilation issue/etc. is causing the problem.