I've also had several employers (from a startup to a FAANG) tell me that an H-1B transfer would be no problem. I haven't actually done it, but I don't feel as mobility-impared as people seem to believe I should.
H-1B transfers are quick (14 days with premium processing), easy and you can file speculatively if you're looking at a couple of potential employers but are afraid it may not go through. Your current employer will not be notified and you do not need to move forward with the transfer if you change your mind.