Non-profits may not run a service better but it does mean they can't be bought (or it's just more complicated to do so). It can be a clear way to signal the intent behind the organisation and restrict what can be done by the company.
In the UK, it's easy to set up a non-profit as a Company Limited by Guarantee, which has no shares. Such companies can still make profits, but they can't necessarily be distributed to Directors. It doesn't prevent them from making profits or spending them any other way.
For a comparison, see past discussions about Post Haven (re: pledges to "never be acquired") . Had they set up as a non-profit from the beginning, the purpose would have been clearer and enshrined in the company's legal structure. Presumably fewer people would have had cause to question the founders' motives.