No, Spirit is/was not a 'small regional'.

You asked if this was caused by or related to bad customer service. This was 100% caused by the increase in jet fuel prices due to the war in Iran. Obviously huge swings in jet fuel prices affect budget carriers more than, say, United or American or Lufthansa or Singapore Airlines, which have many (many) more options when jet fuel prices rise.

Many countries, including many third world countries, have regional airlines. It has nothing to do with America in particular, and the usage of that term is not an American-ism. A good non-American example is Qantas and QantasLink, the latter being a regional airline, and the Aussies refer to it as such.

That really sounds line the US is the only country in the world. Considering the world is bigger, I would call Spirit maybe regional, but not small. Ask some europeans, basically no one will know Spirit - as US people may not know e.g. Wizz.