One advantage to free hosting is obviously the price. Note, if you're running some website that isn't initially supposed to get a lot of traffic, then what's the point of paying? In fact, it might even be possible to use your own domain on the free hosting which makes it even more attractive. Finally, if it turns out at some point you don't want ads or need more resources, then you can just upgrade. Anyhow, can anyone see some downsides to this argument?
When you are getting something for free, you are not getting a product for free, you are making yourself a product
So you're looking at free hosting from the perspective of a freemium model or advertising. If you are showing ads for someone then that's how you are "paying" for the hosting - you could be earning from those ads but you're giving that money away and having no control over the placement, appropriateness etc. If you have intellectual property (ideas, code etc) on a site and you're not paying for it you have zero comeback if the IP gets stolen. If the host goes bust you have zero comeback. If a paid host goes bust they'll normally try to sell their client base to recoup some of their losses and you get uninterrupted service. I fail to see the upside of free hosting for anything other than hobby sites.
There is nothing but upside in using free hosting if you have limited content on the site, say for some hi-metric expired domain you are using for PBN/SEO, that you've already backed up. If the host goes bust or whatever, just put the content on a new or similar host. Not every site requires a full-service host. No payment, minimum upkeep issues, minimum bandwidth issues (since there is limited content or traffic). There are only so many free hosts with decent service, however, so beware of using the hosts for too many domains, as that would leave a footprint (the same IP).
If you value your content, I'd stay away from free hosts. They seldom perform backups of customer data, so you are having to ensure your code and databases are backed up. The time and effort of this alone is worth the $5 a month it would cost. (always take regular backups just in case).