Why are Drupal hosting providers using Solr 3.x instead of 4.x or 5.x?
It probably comes down to LTS and backwards compatibility, though you'd have to ask each organisation for their specific motivation.

Solr 5 is not backwards compatible with 4, 4 is not BC with 3. So if there are users on legacy systems using older versions, upgrading would require all customers to upgrade their schema/integration too. Installing newer versions on new servers might mean customers can't migrate from one server to another, might break the generic model of system deployment, etc, etc.

Some companies will offer newer versions, but you'll probably find the older versions are always available too, otherwise the company is likely to have at least a few pretty irate customers when they switch it off.

Basically, if you tell someone that you're going to support a version of software for the foreseeable future, you have to do that. It's why so many shared hosting companies support obsolete versions of PHP - because the people hosting there have no intention of upgrading their code, but their sites still need to work.

There's no Drupal-specific reason (that I know of) to choose one version of Solr over another, as long as the contrib module you're using has support for that version.