[Bill UK]

Could be the thin wire going to the starter motor that activates the solenoid. One of the problems is the gauge of the wire is not very heavy, although probably ok when new, over the years the copper wire deteriorates, resistance increases due to age and the constant heat changes. This is more noticeable when the engine compartment is hot, the resistance will dramatically increase preventing the starter solenoid engaging, you will normally hear a click sound and that’s about it. You can temporary bypass the thin wire by connecting a 20amp (approx) wire to the spade connector on the starter motor (this is the same place where the thin ignition wire connects to.) switch on the ignition and touch the other end of the wire on the positive battery terminal, the engine should start. Insulate the wire for future use. I did this on my UK Supra before fitting a relay mod. By the way; only the auto box is wired through the ignition switch on the UK models (not sure about the US) preventing the starter operating unless park or neutral has been selected. I think the US models also have a relay that also could be the problem.