I have seen relays that have continuity read across the relays contacts, but will not hold when a load is placed on the magnetic coil inside the relay.
There is a large difference on the load that the horns place on a relay VS what the A/C compressor coil will place on a relay.
Then there can be oxidation problems with the female receptacle where the relay plugs into which cause the relay to not engage or stay engaged.
The fact that your relay stays engaged can be a indication of dirty (oxidized) copper relay terminal connections; after the removal and reinsertion several times has scraped the terminal connections to allow electricity to flow.