"99% of the small web sites which are built around MySQL don't need it."
Likely they are running on a virtual share, and as such as using the cheapest thing available that also supports the web apps they want to use.
If the web app happened to support SQLite, it would still be a better choice to use the hosting provider's MySQL server since it is already configured for backups and likely runs on a separate piece of hardware from the virtual web server. Additionally they are probably using multiple tools, CMS+blog+wiki+forum or some such, and better to just offload all that to the database server.
Even if all these apps supported sqlite, the hosting provider still has to hire a programmer to write code that somehow iterates through all the virtual hosts, finds all the apps running SQLite, and perform backups through the backup API. With MySQL, having all the databases in a central location and a nice community of tools that already handles this sort of thing with a bit of configuration is cheaper.
On the other hand it would be easier on the setup side of the web apps to use SQLite, because no longer will you need to deal with creating the database+permissions+connection strings. Probably the easiest solution is some sort of easily discoverable network service that provides a central backup service, that the host would have for all the SQLite applications to discover and perform backups to.
Just my opinion, but I wouldn't suggest SQLite as the DB of choice for small websites.