I'm pushing for Gitlab (who have an SLA built into their similarly priced offering), but obviously the stakeholders need to decide how we eat this risk.
I use both GitHub and Gitlab for contributing to various open source projects. In my experience Gitlab is far superior. Github is too sluggish and buggy compared to Gitlab.
Does "lighter" really matter in this case? From first-hand experience with Gitea, I can tell you that it most certainly is not more reliable than Gitlab. I've gotten 500 internal server errors from just clicking around the UI too fast while it was running locally on my machine. Constant errors and bugs after running it for about 4 months were ultimately what pushed me to Gitlab.
Don't know anything about Drone CI, but it seems to me like a lot more work to manage two separate systems than it is to manage a single Gitlab instance, where everything is already nicely integrated.
I'm not a sysadmin by any stretch of the imagination, but I was still able to set up Gitlab, Gitlab CI with a bunch of runners, Gitlab Pages, and offsite backups with very little effort on my self-hosted instance. All I had to do was edit one config file to enable the various services (backups were with borg and systemd via the "gitlab-ctl" CLI)
Been running that setup for probably 2 years no with the only outages being external internet/power outages. I'm the only user, so my usage isn't at all going to reflect OP's needs, but it's still more reliability than I ever got from Gitea.