Thank you very much for the quick reply. Would any of the inexpensive scanners be able to show that or would I need a professional grade scanner to check that? I certainly don't mind buying a reasonable scanner, but don't know if it'd be enough to show me the temp reading.

And just to be clear "I" don't necessarily suspect the engine temp sensor, it's just one of the many potential 'solutions' 😆 that I've read in my research. This approach would help me determine if this is the potential culprit. Thanks again.

In situations like this..there are no codes..the ECM probaby is not in Closed mode yet. The only thing is to check the various system for integrity and go to the next....heck you may have to do it more than once..But as far as scanners most basic scanner..(in my mind) would have that data but don't take my word. before buying it would be prudent to see if a particular scanner attracts you download their PDF Manual and read the instructions of operation.
Another suspect is at what RPM does this roughness occur. if you manual jacked up the idle would it go away and smooth out? If it did then suspect the IAC - idle air control is not working properly. On some scanners you can control IAC operations manual to command it to change the RPM to see if it is working...basically your doing what the ECM is doing instead. IAC should be commanded by the ECM to control RPM within the program..under various conditions and loads

Excellent, thank you for the additional feedback. Will check the IAC. With just a couple hundred more RPM it does smooth out, we'll start there and start chipping away at it. Thanks Dion for the guidance....will research scanners too!

Some ideas as a LAST RESORT....the butterfly plate which is the throttle has a stop screw to restrict the throttle plate from going too far closed which would start to gouge the throttle bore. Normally no one touches this set screw but sometimes....yeah sometimes a situation like your presents itself...What I do is adjust the throttle stop so it opens the throttle plate to bleed in air. Now the ECM may not like this and try to close off the IAC some more (if it can) and may even give an error code but if the adjustment is minute - the ECM may not complain..But there it is...not one the factory would endorse - remember you can always undo what you did....

Thanks again, and after trying to find info on how to locate the IAC, it appears that my engine doesn't have one. I have the 2AZ-FE inline 4 in this car, and couldn't find any info on the part nor video of how to locate/clean/replace it. No parts stores had a replacement, and couldn't find it on Toyotas parts page. Finally found that this is a drive-by-wire car and does not have an IAC, TPS, etc.

So it looks like it may be cleaning the throttle body to get things to smooth out? Or would there still be the stop screw like you described in this model?

yeah..looked up your engine model. It appears the IAC is really the ISC or Idle Speed Control system. Check the PDF file.

https://physics.usc.edu/Undergraduate/temporary/corolla2009/etcsi0.pdf

it does have a TPS however built into the throttle body...built in but serviceable...I presume...LOL!

As for the Throttle valve having a STOP screw I am sure it will have one as if it didn't then the plate could scrape the bore of the throttle cylinder. In time for almost all cars..the snapping off of the foot from the gas pedal slams the throttle plate to the stop screw. In time the screw stops becomes smashed in and then technically the throttle incrementally gets closer to the bore durface. NOW not all cars will experience this but is always a possibility. It is just a consideration (however remote) that could exist that is all.

Got it, thanks for sticking with me through these questions, really appreciate the help. Fortunately the car is a manual, so it doesn't stall...just rough idle. I'll do some digging around over the holidays and see if I can resolve the issue. I'll report any findings!