in all fairness, the OP did not suggest that they were going to keep assets history out of control, instead, he was wondering IF they should ignore asset history, so the dangerous idea hasn't foiled in yet -love it though, lol- ?:
…. Well probably the idea is to add the game assets directory into the .gitignore file, but how do I share the rest of the project with the other developers? How are we going to stay in synch with game asset changes?
However, i do agree that they should look into the free Perforce offer for 5 users as suggested by frob (as there only appears to be 4 of them in the OP author's team), this offer will give them peace of mind in the Admin points that you raise about “DIY security and server administration etc…” or setting up the git server yourself, and so they could focus more on the game;
i have to say though i have setup some git servers in the past from scratch, it's fun but i only did it for my personal benefit, not for the team, the team was on bitbucket -dam-
another however is the fact that @babaliaris projected a 4GB payload (DVD size) by the time they reach end-of-game prod, so i'd just make sure Perforce Helix Core offers that much (I have the impression that it is a 1GB offer), I'm happy to stand corrected;
until then ?