If you have 250 engineers (each with a non-trivial salary and total comp) then you can afford hiring or contracting a few good technical writers. That's the solution--there's no slick technology, no methodology, etc. that will make hundreds of already busy and potentially overworked engineers churn out quality technical documentation. Make a case for the value of technical writers, take it to management, build and lead a team to do it...

If you need some ammo, devex ('developer experience') is kind of a hot thing in the agile world right now. Martin Fowler has some good recent writings about how improvements to developer experience (of which technical docs are a huge part) vastly improve the efficiency and output of entire dev teams: https://martinfowler.com/articles/developer-effectiveness.ht...

Spotify apparently has invested a ton of money and resources internally into building a first-class devex platform for their teams: https://backstage.io/ Watch some recent talks about it from Spotify folks for great feedback on how it helped their teams. What they've built is exactly what you describe--self-service, low cognitive overhead to start using and sharing tools across the team.