The "advantage" of a solid D44 over the D35 TTB is relative to what you're doing with it.
The solid axle is stronger and has a slight ground clearance advantage. As mentioned, experience has shown solid axles tend to do good in rockcrawling, so you'd probably want one if that's the kind of use your rig sees.
The TTB setup enjoys advantages over a solid axle too, such as traction on irregular surfaces with the ability of both sides to droop or stuff at once, and increased wheel travel, which has made the TTB a top choice for budget prerunner/desert rigs and the like. The reason the TTB exists though, was it's main advantage, which is ride quality on most roads and surfaces, which is usually a lot nicer than most solid axles. The TTB wasn't made as an improvement over solid axles, it was made to have something in trucks that was still strong and tough enough, but didn't ride like a box of rocks.
I'd say it's more about tire size than anything. The TTB is fine for what it is and what it was designed for. 29-33" tires on the D35 TTB are okay, but after that it might be more trouble than it's worth to keep replacing weak points compared to doing a SAS. Not that you couldn't do a D44 TTB swap and get the strength benefits while keeping the TTB if you want the best of both worlds.
Really though, it's probably more about what's popular and proven and what works. Solid axles have been around for decades, they are the most popular drivetrain choice for 4WD vehicles, and a swap can be done relatively easily and cheaply, at least compared to most other alternatives. So you get something strong, that does the job, and is rather easy to maintain, and theoretically save money in the long run over constantly breaking and fixing something that is not quite up to the task, not to mention time, and spend more time wheeling.
