I had similar symptoms with my 88T although I did have the issues with the coolant overflow. (What year is yours?)
I did chase around for months trying to reslove it with replacing 3 clutchfans, thermostats, and I did discover the rad was some what clogged and had it replaced with a high flow truck core. But that didn't help. It took longer to get to overheating but that's it. I was fighting with a local Toyota dealership who did the last head gasket job trying to get them to admit to there being a problem with some MKIII Supra head gaskets or engines, but by then the head gasket finally let go on No#6.(which is most often the case) And they refused to help.
I ended up rebuilding the engine just 4 month after buying it and 7 months since a Toyota dealership replaced the head gasket. I had her block plained and the head shaved, dipped & Honed to restore her to better than new condition , then used a 2mm HKS stopper HG & ARP bolts. Replaced all the bearings, Rings, Seals, water & oil pumps. Xray'ed the rods, New performance vavles & spring sets.
You name it we replaced it. But when she was reinstalled and she was still over heating.... 7 Months later after the bottom end failure, I had a used 89 engine dropped into her reusing everyting else, and since then it has been amazingly perfect. It has totally different characteristics.
It pulls much harder to 4000rpm without waking the turbo then spools up faster when the turbo kicks in. Has superior economy and a much broader pre turbo power range.
Since then I have come to the opinion that there may be a problem with engines from specific build runs.
P.S.>>>
I'd say you were heading to a BHG.
The first Clutchfans I replaced was aftermarket(lasted 37 minutes) and 2 OEM(that lasted 9 & 17 days)
Your mechanic is correct with his technique BTW. Your Clutchfan is toast and their enlies one of the major reasons for your troubles.
Another observation is that the factory Clutchfans where manufactured with way too little Newtonian oil. I discovered this as being a possability from speaking with Ben from Hamilton ne: of the Toronto Supra Club. He mentioned that he cracked his open and added oil to his to "FIX IT" since I had a few laying around, I cracked open 2 of them and poured the content from one that was totally dead into one that worked intermitantly. It worked.!
Another observation from even more experienced people is that the first CAT is right next to cylander #6 which has the highest failure rate on M7's.
Another question is, does your rad have a complete shroud as well as the belly pan? Both were missing on mine when I bought it so I replaced the shroud but never got around to installing the pan cover. There does seem to be some relation to engine bay air flow and overheating. The shroud alone was a noticable improvement.
Even though this latest engine has been perfect so far(out of an 89), I'm still planning on scrapping the OEM exhaust for a 3" system eliminating the 1st Cat and adding a hood vent just behind the turbo to OUT vent heat from the engine bay.
Another point is to check the rad and clean between it and the airconditioning condenser if you have air. The crap that can build up there is nasty and will fowl the air flow.
P.S.>>>
I also made a home made thermostat mod. I cut off the spring tower assembly and hammered the appature into a smaller diameter leaving a small lip flare that pointed in the direction of the flow to help make it smoother and more focused without adding messy turbulance, This in effect was a wide open "SUMMER" thermostat that was marginally restricted so the coolant wouldn't fly past the rad too fast and still remain in the rad long enough to be effective. Using this mod in the rebuilt engine kept it cool longer and allowed me to cool it down rapidly by just slowing down to 80kph in 4th gear for 2 minutes when she was starting to get hot.
Another quick mod is to either remove the rear engine bay sill gasket to help increase the outflow of heat from the engine bay and increase the flow-through speed at the rad.
BUTTTT!!! The biggest and most effective thing you can do is:
"STAY OFF THE TURBO"!!! I tough myself how to drive her using higher revs and shifting later without tripping the boost(Turbo=Heat) I discovered that this is a much faster way to drive such a heavy car. And besides, The MKIII IS a Grand Touring car at it's best. NOT a 0 to hero drag racing ricer machine. If any MKIII Supra owner thinks they can use her as a red light racer will find out the hard way that they have the wrong machine.
I hope this helps somehow!
P.S.>>>
I didn't include the last bit in specific relation to your over heating issue, but as a general observation I wanted to share with the general population since this would be just one reason for BHG. Miss use!
Appologies for the length of this post.
P.S.S.>>>
I still adore my MKIII and am still sticking with moding her to suit my taste and needs for speed...