A high limit trip will generally cycle the furnace off for a time period. Somewhere between 3-5 off cycles will result in lockout and the furnace will stay off. The control for this is more likely to reside in the furnace board vs an aftermarket ui.
This results in higher efficiency and better steady state comfort as the heat is on for always 18+ hours and often 22+ hours per day. It’s hard to say with confidence on the precision, but it seems like this is saving 8-10% in gas (in therms) vs last year when I had a constant flow temp (that I’d adjust manually every so often to try to match but where I’d have to leave it “too high” or risk a cold house on the coldest days).
The exhaust temps in the stack reach steady-state far before even a 30 minute run would be finished.
This indicates you have a problem with temperature rise across the heat exchanger, probably from inadequate airflow. Your ducts may be sized too small/furnace too large, fan speed set too low, plugged filter, etc. You may be able to mitigate the problem by increasing fan speed, limiting heat by using stage 1 only, putting in less restrictive filter, add additional cold air return, etc.
Usually a failed limit switch means cycling and a bigger problem with airflow. However I have seen a high temperature limit switch fail without cycling - fell apart with age (40 year old furnace).
It took an extended run to get the house back up to temperature, but it had no problem with a correctly functioning switch.