Potentially yes, but depends very much on the privacy policy and data handling promises being made.
I think the instinct to distrust big companies is at least partly because many of them have already proven not to be good stewards of data which when combined with their scale has more worrisome implications.
With a smaller/newer player, at least there’s some hope that they’re not capable of the same harms at a smaller scale, and in some cases may market themselves specifically as a more private alternative.
Whether or not this turns out to be true in practice and over the long run is another thing.
Hope ain't the same thing as trust, though. A small player would need to make a pretty significant effort to suggest they wouldn't abuse your usage-patterns.