What is the per-gene score in BUSCO output?
You are correct, it is the score of the candidate protein against the BUSCO profile hmm assessed by hmmer. If it passes a threshold, it is validated as being that BUSCO gene. The threshold is defined for each profile based on the score of the species used to produce the profile. It can be found in name_odb10/scores_cutoff. And yes, the higher the better.