I’ve recently started hearing this form of pedantry, but I’m not sure I understand it. “Performant” seems to be defined by the New Oxford American dictionary as “performing well” and claims it is mostly used in computing contexts. What’s the issue with saying “most performant”?

It's vague. How does it "perform well"? Does it mean "faster"? "smaller"? Why not use a word that's not only more specific and well-known, but faster to read and pronounce?

I believe that’s a separate issue as “best-performing” isn’t any less vague.

FWIW, to me performant has connotations of conformant, so performant implies "performing to spec". I also think it implies in general doing what it's supposed to. Maybe that impression is incorrect but it is my impression.