The existence of a finite trivialising cover is a less stringent condition than one would expect: see https://mathoverflow.net/questions/94479/does-every-vector-bundle-allow-a-finite-trivialization-cover

(Sorry for the commentlike answer but it seems that by migrating to SE I lost reputation points, so I have not enough of them to properly comment)

Edit: however, the uniform norm control over the cover might be an issue when $X$ is not compact, so my comment is really not that helpful I guess.