As the founder of one of the sites on this network, who has put an insane amount of time into building that community of 6000+ users, I want to escape from SE just as much as anyone else, but certainly there's disadvantages. Nothing below should be interpreted as me saying it's a bad idea to exit. It's just my honest opinion.
SE is constantly getting DDoS attacks. A security team helps us deal with those, and helps ensure that our passwords and private data (e.g. the IP addresses from which you accessed the site each time, which can tell hackers your address, your choices of hotels and locations of travel, the family members you visit as long as you view MathOverflow from your phone while there, etc.) are somewhat safe. Having SE-level security will likely not be cheap. Here's a list of security vulnerabilities that were found by non-MO users of SE over the years.
SE currently uses 600,000 web sockets, 3 elasticsearch servers (128GB of RAM each), two redis servers (256GB of RAM each), about 2 SQL servers (768GB of RAM each), 9 web servers (64GB of RAM each), and deals with a transfer of 55 TB of data per month. MO will not need that much, but running and maintaining servers has a cost.
User participation would drop
Let's look at some of the other success (actually "lack of success") stories for Q/A communities that are not part of the SE network.
PhysicsOverflow. Notice the lack of usage, the archaic aesthetics, and the fact that the site looks completely messed up when you try to view it from a phone rather than a desktop. I say this with the utmost respect for the founders of the site, knowing that starting one of the SE sites and getting it through Area51 was the hardest thing I've ever done in my life, and that I'd still probably prefer having gone through what I did versus trying to do everything on my own.
Codidact. Notice that nearly no one uses it, including the main "founding face" and popularizer of the community, which is Monica Cellio (until recently when this strike action started, she hadn't posted there for a long time, despite probably being the one with the most vested interest in growing that community). Also be aware that the vast majority of the content there is simply copied and pasted from SE thanks to the creative commons license which allows them to do that. I say this again with the utmost admiration for what the Codidact founders have accomplished and for their vision, which I support enthusiastically (I was in fact one of the early contributors to design elements that are still in place). Another problem with this community is that the main founders (Monica Cellio, ArtOfCode, Mithrandir, Mithical, luap/MEE) have in my opinion too much in common with the people currently running SE: 100% of my conversations with Monica about Codidact have been pleasant, but I have had mostly negative experiences with the other four.
TopAnswers. This community is so unknown that I don't even know what to say here, other than the fact that I occasionally see this site mentioned as an alternative to SE, for example here.
Materials Science Community Discourse. This is probably the closest to what the MO community would want to do. it's powered largely by Discourse which is also free and open-source like the Q/A engines in PhysicsOverflow and Codidact, but unlike those two, it's much more widely used (and one of the founders is actually Jeff Atwood, who also founded StackOverflow!). However I can already see that the Materials Science Community Discourse website has a big banner saying that it's not supported on my browser (not a problem with SE). Also, it is a "traditional forum" with no voting, comments, or any of the other things that make SO/MO/SE the way it is.