Note that the GitLab Web UI is no longer the only possibility to trigger a blame.
There is now (August 2019, GitLab 12.2) a GitLab blame API (from Oleg Zubchenko)
In GitLab 12.2, the new Blame API allows this information to be retrieved directly from GitLab, without needing to checkout the repository.
This is helpful for scripting and automation, based on the people who’ve recently changed the file.
See documentation and issue.
I suspect this will have the same limits you have seen with your large repository, though.
You can see "git blame" with GitLab 14.7 (January 2022), but only for code search results, in the non-free tiers:
Go to Git blame from code search results
Users often want to understand more about code search results, such as when was a file changed or by whom.
With GitLab 14.7, users can easily answer these questions with fewer clicks by using the View blame link in Global Search results.
This change adds another link next to lines of code when hovering over results from a code search.

See Documentation and Issue.
GitLab 16.8 (January 2024) adds:
View blame information directly in the file page
In previous versions of GitLab, viewing file blame required you to access a different page. Now you can view the file blame information directly from the file page.

See Documentation and Epic.
git blamelocally?Too many changes. To preserve performance only 100 of 131 files displayed.It looks like the problem was that the person's changes I was trying to view was usinggit add .instead ofgit add -u. Sorry I'm not able to provide a specific error code, but I think that's it.