Well, from my experience as the person who "babysits Jenkins" unless you appoint an actual owner for Jenkins, you'll get a real tragedy of the commons with it. Devops sounds cool and all but most devs want to write cool code: usually product features, not infrastructure stuff, so everyone will do CI/CD related work half-heartedly.
So in my obviously biased opinion, if your team is big enough, either get an actual build/release engineer/devops guy or appoint one of the devs as an owner for this, otherwise you won't get the best results.
$12K/month for a much (I mean, it's like going from cvs to git) better solution in that situation should be seriously evaluated; perhaps you could give every team in that org their own dashboards and eliminate/distribute the whole release engeering org. $12K/year, which is the actual TeamCity pricing, is a no-brainer, which unfortunately most teams are way too cheap to pay for, headcount looks so much better. Fortunately at least cloud based CIs are starting to seriously take off, hopefully I never have to deal with a home-grown Jenkins deployment again.