Not necessarily. As I explained above, most successful acquisitions are stock sales, in which case the acquiring company now owns the startup (they hold the shares). The startup is still a separate entity at this point.

Google is known for just merging the acquired startups into their product line (and/or killing them), but it's not a hard rule that all acquisitions are mergers.

For example, AFAIK Livestream is still a subsidiary of Vimeo (ie wholly owned, but separate): https://en.wikipedia.org/wiki/Vimeo_Livestream

So Livestream can be profitable or not, separately from whether its acquirer is.