8600 tuner and Tuner Genius XL

Great question, and the 3:1 vs 10:1 matching range difference is really just the starting point of the story here. The deeper issue is where the impedance is being measured.

The feedline between the antenna and the tuner acts as an impedance transformer. Due to the standing wave pattern on a mismatched line, the impedance seen at the tuner input varies continuously along the feedline as a function of the electrical length at that frequency. A center-fed Zepp below 7 MHz can present a very high impedance at the feedpoint, and depending on the coax length, that impedance can transform to something relatively benign by the time it reaches the shack (say, 200-400 ohms) or it can arrive as something extreme (several kilohms, or a low impedance with significant reactive component). The TGXL is rated up to 10:1 SWR, but if the transformed impedance at the tuner input falls outside that window, it simply can't find a solution. Changing the feedline length by even a quarter wavelength at the operating frequency can completely change what the tuner sees, and sometimes that's the fastest fix.

The 8600's internal tuner, being right at the transceiver output, sees the full coax run as the "antenna" and the pi-network topology is inherently well-suited for matching a 50-ohm source against a wide range of unknown impedances. At 3:1 it's more limited in range than the TGXL, but on some feedline lengths the transformed impedance may simply land closer to 50 ohms and give the internal tuner an easier match on those low bands.

On the common mode question: coax feeding an off-center or unbalanced antenna like a Zepp is almost certainly carrying common mode current on the outer shield. That current flows back toward the shack, and the shield becomes part of the radiating system. What the tuner then "sees" is not just the antenna impedance but the antenna plus the outer shield plus whatever the shack ground looks like. This makes the impedance at the tuner input both unpredictable and frequency-dependent, and can explain why one tuner finds a solution and another doesn't.

To directly answer your question: no, neither the internal 8600 tuner nor the TGXL includes a common mode choke. They are both unbalanced (coax-in, coax-out) devices. A good 1:1 current-mode choke (a.k.a. choke balun) at the antenna feedpoint, wound on a high-permeability mix-31 or mix-43 toroid, would block common mode current from traveling down the outside of the shield. This would give both tuners a cleaner, more predictable load to work with and would likely reduce the variability you're seeing across frequencies. On 160/80/40 you want a choke with high choking impedance (ideally >5 kilohms), which usually means multiple turns on a large core.

Worth trying: add a feedline choke and also experiment with feedline length on the bands where the TGXL is failing. A half wavelength change in coax length at the problem frequency will present essentially the same impedance the feedpoint does, while a quarter wave change will invert it. You may find a length that puts both tuners in a happy place.