The problem is still concurrency. If the application, in this case the git CLI, is not thread-safe, then you could run into unsafe data mutations causing data corruption.

Imagine two pushes that come in around the same time. S3 doesn't have the locking mechanics you need to ensure that only one of those writes win and the other fails. You need to upload the individual commits (which can be isolated except in the case of multiple writes with a partial write), but then the branch and HEAD pointers have to be serialized. I've done a lot of work with distributed systems in AWS and my experience has always been to layer something like DynamoDB on-top of S3 to provide this.

My knowledge of at least one distributed Git system was that they had a fleet of Git data servers and a router on front. The router would fan it out to 3 or more data servers where each data server would store copies. But ultimately there's some application logic that is aware of the underlying filesystem and manages locking.

GitHub shares some of their architecture here: https://github.blog/engineering/infrastructure/stretching-spokes/