If there’s no codes being thrown up then start with the obvious things first as there’s that many sensors in cars now something should have thrown a code up, check for routine service parts like air filter plugs oil filter, check EGR and DPF if its diesel, just to rule out any blockages that may be causing it before you start taking apart your engine or transmission. also if the temperature shot up double check the thermostat, I know this wouldn’t cause high idle but check its in good condition as there cheap to replace.
bad timing issues would cause lower than normal MPG, burn oil, loss of performance, noisier engine especially if its got a timing chain, it may struggle to start or randomly stall, misfire, rough idling high or low or fluctuating.
if its your transmission the your issues would include loud whining and clunking noises, lack or response, maybe a burning smell from overheating fluid or leaking fluid, transmission is noisier than usual in neutral.
I think your car has a timing chain rather than belt so there usually isn’t a service interval for replacement but would be worth checking with a garage to see if they can check the condition of it anyway as its a costly repair if it goes wrong, if its stretched or jumped a tooth then it could cause some of these issues but not usually revving high as that. guessing its an automatic transmission, check for any common issues or see if owners are having to service parts around the same time.
try taking it to another garage and get a second opinion if you still think it may be the transmission, by the sound of the symptoms your explaining I would be swaying more towards gearbox/clutch issues as this is what’s happened to me in the past and its always turned out to be them. not sure about the temperature spike though, has that only happened the once when the revs shot up that high? is the clutch slipping at all?