Would False Positive be a good name for a medical screening service?

It would be a logical name if its customers were technicians familiar with LLMs, and not end businesses and consumers. Which is why Ford wasn't named Internal Combustion Engine, Apple wasn't named Graphics Processing Unit, etc.

Generative Pretrained Transformer is also a terrible brand name but it doesn’t seem to matter.

Because the name isn't Generative Pretrained Transformer, it's GPT.

Perplexity sounds no weirder to me for a tech product than than Google or Yahoo or Apple.

Those are all short and easy to say. Very obvious difference between them and a long complicated word like Perplexity.