You still haven't justified this:
But the residual heat part tends to be far smaller than the lag part
but the discussion has been about controlling room temperatures, not boiler temperatures.
Can you tell me another reason for investing in more advanced controls than hoping to use less gas and save a few bob? If you want to eliminate unnecessary gas use from too high a room temp, simply turn it down a bit.
But that is a completely different subject; the OP is concerned with maintaining a constant comfortable temperature.
I've been around considerably longer than you and can remember the times when mum/dad would be forever turning the stat down because the room was too hot and not long after turning it down because it was too cold. The stat may have been set to 20C but the room still heated up to about 22C,
due to the residual heat in the rads. The stat was turned down until it clicked (about 18 or 19C) but the room cooled down below that due to the hysteresis in a bimetallic sensor.
It's a well documented fact that if the temperature is maintained with close limits, a lower temperature can be tolerated.
You have introduced a lot of irrelevant information about the efficiency of SE and HE boilers, but no concrete evidence to justify your earlier assertions.
Since you have never been a heating engineer, I don't expect you to understand what I explained, but even you can find it on Google if you read enough so you can absorb it in small steps.
I will treat that with the contempt it deserves.
If you want evidence that a TPI controller works, I can provide it.
When I moved to my current house, about five and a half years ago it had the traditional electro-mechanical stat with anticipator. The first years gas consumptions was about 30,000kWh. I then installed a Honeywell CM67 followed by a CM927. The consumption has gone down to about 24,000kWh; a reduction of 20%.
And before you reply that the saving will be due to it being warmer in the winter, think again; winters have actually been getting colder. Look up the degree-day data for your area.