Are systems like Git robust enough so that if questions arises about a paper's priority, it would prove or substantiate a claim to have a VCS log?
It sort of depends on what you want to timestamp, and who you want to convince with the data, but there's some important points to keep in mind.
If it's a locally-kept repository, then it simply cannot be used to establish any sort of priority, because those timestamps can be forged. Git is simply not robust enough (depending as it does e.g. on your system clock) to be a silver bullet; it can be made part of a case (analogously e.g. to timestamped lab books) but it is not enough by itself.
On the other hand, you can use a version-control repository hosted by a third party, such as a private repository on GitHub or Bitbucket. Here you need to make sure that you push regularly to the remote repo, because what really matters is the timestamp on the push and not the commit (again, because the locally-generated timestamps can be forged); you also need to make sure that the third party actually keeps timestamped logs of said uploads.
This is in principle usable, but whether it will be sufficient or not will depend on what you want to do with it. Ultimately it will come down to whether the people you're trying to convince can be persuaded to trust the third party's logs and their neutrality. (Note, also, that what matters here is not the robustness of git that matters - it is the reliability of the third party instead.)
Depending on your field and its community, having written the paper first may not actually matter - quite often, a lot more weight can be put (rightly or wrongly) on being the first to publish the work.
There is a lot of field-to-field variability, but if someone else publishes work that is essentially equivalent to yours, then at the very least you should expect a steep uphill climb to get a journal to publish your manuscript, even if you can prove you had the text just about completed before the others even submitted. I will leave these discussions aside, though, under the assumption that you've already thought through them or will do so before acting.
I understand your question in the following way:
You are asking if a git repository can be used to add a "timestamp" to some idea, so that in case multiple researchers come with the same idea, you can prove that you were the first (whenever this is actually true).
Unfortunately, dates in a git repository can easily be forged. See here for a stackexchange thread documenting this.
Trusted time-stamping service
What you are looking for is a provider of trusted time-stamping.
Git can not — or rather: should not — be used for that, because git is not in any way meant to perform that sort of service. For git to be used this way, you would have to prove that the git repository is accurate when it comes to time, accurate about who is submitting to it, and tamper-proof when it comes to commits.
And sure, you may assume that this is the case with any of the big third-party providers, like github. And sure, in a civil case, where a preponderance of the evidence is enough (i.e. evidence need only be more likely than not likely, instead of as in a criminal trial where they must be proven beyond reasonable doubt), then this might very well fly.
But why bother with re-purposing git when there are dedicated providers for this very type of service? There are both free and paid providers. I would personally take someone way more seriously when they use a proper service for this compared to someone that re-purpose something that is meant for completely different things.