"Bad starter or relay" is a common throw-away line I interpret as "known bad engine". Agree on a price (probably <$3K), with the condition the engine checks out.
Before paying anything and with them there, run the engine. If truly a bad relay, just jumper across it (make sure it is in Park). Use jumper cables from your battery if needed. If you can't measure compression on the spot, pull the breather and the PCV and look for noticeable blow-by like regular puffing (bad rings). Look for oil in the coolant (bad head gasket). If the engine won't start or is bad, knock the price down another $800. Be prepared to walk away.