The actual warranty - an insurance product really - is actually provided by a specialist third party which is not actually Tesco.
Tesco apply their brand to it though and inevitabley take a cut, in the same way they would with other Tesco financial services, eg credit cards, pet insurance, car insurance, etc.
Having the Tesco brand associated with it gives me a little more confidence.
I don't recall the position regarding FSH - I didn't have to think about it as I do have FMBSH - a quick call should sort that though.
I contacted them when taking out my policy to check that superchargers are covered - turbochargers are mentioned specifically, but not SC - they quickly checked with the underwriters, and IIRC sent me an email to confirm.