i would be more concerned with the link network being devalued than the hosting service's business model. since you would likely be on a shared IP there's always the risk of being in a "bad neighborhood" which might count against you. for example, you might end up on the same IP as a google-identified unnatural link network.
Free web hosting has a long history of problems. Some of them have blocked googlebot in the past to save on bandwidth. Other free hosting sites have added hidden links to client sites. Most of the time free web hosting does not have a problem but why risk it. If these sites have real value then you should invest a few dollars for real hosting.
ps I've been using this link tactic in Google for over 10 years, they know about it and have become pretty fast at spotting it and devaluing it. I'm not saying it can't be done just saying most people that do this today are not happy with the results.
since you would likely be on a shared IP there's always the risk of being in a "bad neighborhood" which might count against you.
This is true of paid shared hosting, too. You just have to check out the neighborhood and pick a host which doesn't allow gambling, adult sites, etc. And some free sites are more likely not to host that stuff than some of the really cheap shared hosting plans. So if you're on a really tight budget, free hosting could actually be your best bet.
OTOH, look for problems with slow page loading and outages. Free hosts aren't always reliable for those things, and from my own experience with unreliable hosts, I do think Google can get very confused by downtime and slow loading speeds. (It's the kind of issue that, once resolves, usually gets you right back to where you should be in Google.)
You could definitely start free, see if Google seems at all interested, and THEN find decent paid shared hosts once the sites are producing enough to pay for it.