Trane and Lennox are good brands, but not the only ones worth considering. A lot of equipment is similar nowadays with respect to available features, performance, reliability, etc. You have to be comparing apples to apples, however.

The Trane 20i is a nice system with two compressors for better staging between high and low (around 50% capacity on low vs. 75% for the Lennox).

Lennox doesn't "use" ADP coils. The dealer is pairing it with an ADP coil likely to boost the efficiency of the overall systme (if trying to get tax credit qualifications). Probably just a misconception there.

Trane uses aluminum for most of their coils (though not in air handlers with the current production, but this is supposed to be changing soon). Post a model number. The Trane dealer may be using an Aspen third-party coil like the Lennox dealer. I would stick to Trane and Lennox OEM coils if possible. Many Trane coils are aluminum (4TXC model is, but not sure about all others), but that doesn't mean they won't or can't leak. I do think the aluminum is more reliable in terms of leaking than copper (due to corrosion), but both can leak, and most brands provide a 10 year warranty now to protect you somewhat.

I'm replacing my ADP coil after two years of operation. I have a small leak in one of the capillary tubes. When I spoke with a Trane authorized dealer I was informed of two things. First the ADP coil does not carry the same warranty as Trane 1-5yrs in my case vs. 5-10 for trane. Second, even though ADP is going to replace the coil they will not warranty it beyond the date of the original start-up. ADP's clock starts running 1yr after manufacturing. They don't keep records of when they were sold or who they sold them to. This is probably not the case it's their customer service center passing this along. So as I would see it if your Heat and AC guy is pairing an ADP coil with your AC it's about cost. Here in Ga many companies working for builders rig these combos to save the builder money and to keep a job with them. Save yourself the headache and go with all trane parts because you aren't going to get the warranty with ADP. Most importantly our service men are receiving calls on these all the time.
ADP builds them to order now so they don't have as much standing inventory as they once did, but consider this. They build a coil in the hay-day of building and it sits on a shelf for 6 mos. The builder has a system installed around that same time but doesn't sell the house and 18 mos later the bank owns it. Someone comes along buys the house from the bank and has the system turned on and the coil is already out its 1yr warranty. The Trane furnace is still in warranty but the coil is not. Just a thought.