I can see the stat opening on the gauge.
Does that mean you can see that it stops rising for awhile when the thermostat opens? That seems different to me than "Just keeps going".
Is the cardboard air deflector in place? Early 3A didn't have them, but they are definitely a recommended retrofit. But the main point is that it hasn't slipped and fallen in front of the radiator.
Check the fan belt; the lower generator mount often comes apart which can let the belt slip. Also check the groove in the water pump pulley to see if the belt has polished the bottom of the groove. If so, the belt may have worn until it bottoms in the groove, which allows it to slip no matter how tight it is.
Hook up a timing light and check that the ignition advance is working. The front pulley should have a hole that marks TDC when it lines up with the pointer on the timing cover; but sometimes the pulley gets assembled wrong. At idle, the hole should be around 1/4" from the pointer when viewed under the light. Rev the engine up slowly and let it back down, the mark should move smoothly in response. Only a rough check of course, but worth doing. If the mark doesn't move, or starts jumping around at any point, the distributor needs rebuilt.
Although not common, it is possible for the WP to fail. It's easy enough to remove to check.
I've heard of people getting the radiator out without pulling the apron, but pulling the cylinder head sounds like more trouble than it's worth to me. I've always just taken the apron off; which isn't as bad as it seems as long as it's been off recently. A 7/16" GearWrench with an offset head makes the job go easier.
Unfortunately, the radiator is often the problem; and may not be obviously bad. The original radiator in my previous 59 TR3A apparently developed corrosion in the joints between the tubes and fins, so it just had lousy thermal efficiency even though the tubes were open and clear. I fought with it for a long time before finally pulling the apron and having the radiator recored. All my cooling problems vanished like magic (of course by then I had tried several improvements).
Fast forward 10 or 12 years; I removed that radiator from the wreck (along with the electric fan), had a radiator shop check it out, and installed them both in my current TR3. Which promptly overheated worse than it had with the original radiator and mechanical fan! Eventually, when I insisted that the radiator shop rod it out (even though they said it flowed fine, twice); they discovered that the tubes were coated inside with "mud", probably a mixture of rust flakes and stop-leak. (I always had a problem with the upper tank cracking around the extension.)
Once again, a recore made all the problems vanish. With the electic fan and a few other upgrades, the needle rises to the 185 mark and stays between the center and the edge of the '5' where the fan comes on.