That's an interesting idea although the devil is probably in details: heating the basement might mean heating the rooms less which might offset any benefits from warmer floors. I suspect I am losing a lot more heat through walls that through the floors.
How long are your heat cycles? It’s not a bad thing for the furnace to run a little longer, in fact too short of a run cycle can become problematic in some situations.
By keeping your feet warm you can offset some of the cold “feeling” you get as you walk around. This more applies to those who walk around bare footed rather than someone who always wears slippers. I don’t doubt for a moment that you feel colder on the outside rooms, this is common because of the heat loss through the walls and windows. The warm floor thing would help to slightly reduce that (if your floors were noticeably cold all the time).
Have you used a thermometer to measure the air temp difference between the rooms that feel cold and the room with the thermostat? What is the difference between the rooms?
I’m going by the information you mentioned earlier, since your unit is able to satisfy the heat demand I’m going to assume that your furnace is not undersized. If it ran constantly and just couldn’t keep up then you would have an issue. It sounds like what you have is a balancing issue, or an insulation issue. You said you upgraded your insulation already so that leaves us with balancing. The room that has your thermostat is the room that controls the temperature of the whole house. When it comes up to temperature then the thermostat thinks the whole house is up to temperature. If the rest of the house is still cold then the room with the thermostat has too much air going into it. It’s heating up too quickly. Reduce the air into that room so that it doesn’t heat up as quickly and therefore giving the other rooms a chance to warm up too.
If all the rooms are comfortable, and at the same temperature at the end of the heat cycle but cool down exceptionally quickly then you still have a heat loss issue or drafts somewhere that is causing issues.
The heat naturally wants to rise, so you need to protect the heat that stays low. You have an un heated basement and that is a heat drain. It’s allowing the heat that is low to transfer through the floor. This is taken into consideration when heat load calculations are preformed. Additional heat is required to be supplied to a room if the floor is not insulated or over an un heated area. If you can not add additional CFM to each of those rooms then heating the space below would reduce the required CFM and achieve similar results (within limitations). Some of that heat below is going to travel through your floor and into the space above so it will reduce the speed at which those upper rooms cool down. If lack of insulation is still an issue then this won’t be a substitute for it but it sounds like you’ve fixed that previously so I’m just pointing at the next smoking gun that I see from where I’m sitting. I don’t recall exactly where you said you added insulation though...
Of course, you are my eyes on this so if I’m misunderstanding something or not seeing the full picture then you would be the best one to make that call. All I can do is give you my opinion from the information that I’ve been given. I’m still going to recommend a heating contractor go there to preform tests and calculations because that is ultimately going to give you the best answers to the problems that you are having, providing that you get a technician capable of performing those calculations and tests correctly... that may be your biggest challenge. Be sure to let the service company know exactly what you want done so they send the right tech for the job, and not the next available apprentice.
Im really curious how long the furnace runs on each heat cycle and how long it’s off between cycles. If you could time those and report back it could help us know if you have something vastly abnormal. Record the outside temperature when you time it so I have a rough idea on what it’s working against. These measurements should be taken on its natural run cycles without increasing the temperature setting to make it turn on. Also, you should not be standing beside the thermostat while you time this, so body heat doesn’t affect the results.