This is a REALLY GOOD question and I think that I can help but first we need to clear up some things that I think that you are misunderstanding.
First film needs to be developed/processed which creates negatives. Then the negatives need to be printed.
Do not confuse developing with printing, they are two separate processes.
Any lab that develops film can develop it thus creating negatives but how they print and what they print on is what your question is centered on. Your question has nothing to do with developing.
Modern minilabs do not make traditional, darkroom style optical prints on silver gelatin paper. I can think of none who do it this way anymore aside from a few very small custom shops such as Full Circle in Baltimore, MD. This printing method is largely reserved for the do-it-yourselfers who are printing in personal, school or public darkrooms.
The vast majority of labs make prints using inkjet/pigment prints. And for color photography, regardless of if you shot on film or digital, this is really the accepted standard. You probably won’t find ANY lab who will make optical color prints.
That being said, you can send your digital files or negatives to labs such as Ilford where they can make silver gelatin prints using a modern LCD enlarger. https://www.ilfordlab-us.com/
But you realise, silver gelatin is for b&w only right? Also glossy and matte are only finishes to a paper, not a type of paper in itself. There are also glossy and matte silver gelatin papers.
Personally, all photographic prints that I get made for my home or to sell at galleries and to clients are printed at Ilford in silver gelatin if they’re b&w or MPIX.com if they’re color, which are just high end inkjet prints available in various finishes. I am also a film photographer working mainly in 35mm, usually b&w these days. And I simply send these sites my files for prints, I have not made an optical print in a darkroom for many years, but that is certainly an option too if you want to get to that level of quality; build yourself a darkroom!
It’s good that you’re seeking out prints that are of a higher quality and have more character than what you are currently receiving. Don’t stop exploring your photographic options!
I hope you find what you’re looking for. Ilford Lab is probably a really good start. Happy shooting and please remember to choose a Best Answer.