What I have:
Burnham gas furnace, hot water, model P205 WI (it's 30 years
old but still in good condition)
There's no stack vent (it had an old HyTemp which I've
disconnected electrically and locked open).
No fans of any kind.
Electric circulating pump.
House is old (but not large) and has big cast-iron radiators.

I was having some problems (furnace wouldn't start
correctly), had my plumber/heating guy come over. Checked
the pilot/ignition, cleaned the sensor.
He installed a new thermostat (old one must have been 40+
years old).
The new thermostat is a Honeywell "round" T87K1007.
There are only two wires from the furnace that connect to it.
No fans, no air conditioning.

The furnace still wasn't right afterwards, but I
investigated on my own and came to the conclusion that the
intermittent ignition module (old Honeywell S86H) had
something inside that was failing. I replaced it with a new
Honeywell "universal" module (S8610U3009/U), and that seems
to have solved the problems with the furnace.

But I find that the new thermostat is quirky.
The "room temperature" indication (bottom pointer) seems to
have no relationship to the "desired temp" setting (top
pointer that you set by rotating the dial).

I checked the thermostat's DIP switches (it has these
instead of the "predictor" that the old one has), and found
one that wasn't set according to the manual, and reset it.

But... I've come to the conclusion that the thermostat
doesn't do what I want it to do.
Please keep reading.

I understand that a thermostat is intended to automatically
maintain room temperature within a slight variation from the
desired target setting. Perhaps within 1 degree +/-?

The result is that the furnace cycles on/off too much for
what I want.

What I've been doing (it's still cold at night here in
southern New England yet) is setting the thermostat up
higher than where I want, letting the furnace run for an
hour, then turning the thermostat down (and furnace off)
until it gets a little cooler than I like.

I can repeat this two or three times a day, and that's all I
need.

So....
I know there are "smart thermostats" one can buy, but I
still don't know if they can do what I want. Which is:
- set a high temp limit (furnace shuts off)
- set a low temp limit (furnace turns on)
Say, set my low for 64 degrees and my high for 70 degrees.
Or something like that.

Are there any thermostats out that that work this way?
Or do they ALL "cycle within" a smaller range of temperature
change?