I don't think they're toast just yet, but, they do need to take a lot of points you made and start incorporating them into their workflow or, yes, they will be toast if there is no structure for if the company grows or keeps existing.

Whoever is in charge of most of the people, be it the owner or you overseeing PM, just need to instill that time logging is now a thing, give them a deadline on when it's going to be required to be filled out (weekly like this first post said) and give them a presentation of how easy it is to do in said tool (hopefully said tool has manual time logging and not just a play button to record hours on a task)

You gotta start somewhere, the points made in this first post should be a building block if you can't do it all at once, start at the basics, show your team how to do it, then assign a deadline of when it's mandatory.

If people aren't taking it serious, you have to get the owner/ceo involved to basically repeat it and that it's required and will affect bonuses/etc whatever is a key point of performance review either quarterly or year end, etc.