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Well, there are many variables there.
Every brands builder grade, base model units, are prone to early failure and annoying problems.
Of the three you mentioned, the higher end units for all three brands are good systems.
Carrier and trane make excellent AC units. Carrier makes excellent heat pumps
Trane's gas furnaces often tend to have issues. Lennox makes a good unit but finding parts is not quite as easy as for the other two brands.
When you are asking personal opinions of HVAC technicians, you can ask ten techs and you will get eleven different answers.
Anyone who is a dealer for a specific brand is always going to say that brand is best.
If it were an AC or heat pump, and I was installing it in my home, I would go with Carrier.
(Being limited to the three brands you listed)
If it were a gas furnace, I would go with a lennox, or one of carriers units with a tubular stainless steel heat exchanger. The stamped steel and lower grade steel tubular HEs are far more likely to fail prematurely.
You can find carrier parts nearly anywhere, and many of them can be replaced with generic parts found on the shelf of every HVAC supply house in the country.
Trane has a few more parts that are Trane exclusive. (which means they cost a lot more and take extra steps to get them) Lennox can be even more of a challenge to find parts for. Now some would say that the lennox breaks less often so you wont need parts, but everything man made breaks eventually.
In my personal opinion, I would rather have a system that I can find parts for very quickly, very easily, and at decent prices. In my experience, carrier has done a better job of that.
I hope this is helpfull and not just more confusing
If you have any further questions please let me know.