It really depends, if your setup is docker based (as OP's seems to be), adding something outside is not a good solution. I am talking for example about traefik or caddy with docker plugin.
By versioning I meant that when you do a push to master, you can have a release which produces a new image. This makes it IMHO simpler than having just git and local files.
I really don't see the complexity added, I do gain isolation (sure, static sites have tiny attack surfaces), easy portability (if I want to move machine it's one command), neat organization (no local fs paths to manage essentially), and the overhead is a 3 lines Dockerfile and a couple of MB needed to duplicate a webserver binary.
Of course it is a matter of preference, but I don't see the cons honestly.