Agree with you. Each tire pressure sensor inside the wheel, have built-in life battery, temperature and ID registered.... This ID registered is where the Tire pressure module can see the frequent signals, then computer can ID the tires on 4 corners or spare tire. What if you rotate the tires? Corrupted TPMS computer and set TPMS warning light on. Many vehicles (early model year) have no trouble with TPMS lights after tire rotated.
Most time the flat tire get air pressure proper, it will turn TPMS light off after driving few miles.
Toyota/Lexus vehicles make me miserable, you have to check a spare tire in the trunk or under the vehicle because the TPMS computer will see 5 tires. What if you open a trunk lid and see full of hoarded stuffs inside, will you bail the stuffs out to access the spare tire? Me? No way, not worth to check it unless customer request for check spare tire air pressure. Many times I see TPMS lights on as the vehicles brought in for oil services. Ignore them but I put air in the tires then TPMS lights gone or stead on because customers are aware of TPMS situations before.