How do I search for a thermostat that has a delay before it turns on heating, rather than instantly turning on once the temperature drops by 1 degree?
If you only have a 2 wire thermostat it’s not the thermostat that needs to be changed. A “delay ON” relay would allow the firebox to get to temp by a set time but you don’t want wait for very long or there will be damage to the heat exchanger and trip the high temp safety (usually requires manual reset).
For gas furnaces that are loud on start up I have found this to be because the burners are dirty the chamber fills with gas and finally there is enough and whoosh it lights, this can even blow the pilot out in some cases.
Cleaning the burners or having them cleaned will usually eliminate the noise as the gas is lit as it enters (I have serviced quite a few gas furnaces and this is a common issue).
You may be able to service your furnace, open combustion usually have multiple burners that just lay in place with the furnace turned off and the gas off, reach in and lift the burner and slide to the Back they usually come out quite easily.
Next take a wire brush to the top of the burner (the trick is to point the burner down and wire brush then the rust falls on the ground) after brushing a light tap or 2 until no more crud falls out. Repeat with each burner.
The burner closest to the pilot is the most important and the pilot may be screwed to this burner. Remove screws and remove with some brands you have to remove one to the side because the pilot is in the way for direct removal, then put everything back turn on the power and gas and light the furnace if it has a pilot.
It will probably be much quieter lighting now.
If you have a hot surface igniter the same process has worked but make sure not to touch the igniter any finger oil can cause the hot surface element to crack and then it will need to be replaced.
For thermostats  that can adjust their “dead band” or “span” most are programmable that have settings +- for on or off like winter -1 for on and +2 for off I change mine for summer -2 for off +1 for off. Because this heels best to me.