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These Tundras have a craptastic Electronic Limited Slip rear end not true LSD the computer applies the rear brakes on the side that is slipping. Honestly the worst thing about these trucks they should have a true LSD.
While the electronic LSD works it is very loud when the brakes are actuating and you actually have to hit a button to activate.

I’d like to get a true LSD installed but the install is pretty involved and far beyond my abilities so the $1k in labor alone is tough.

Regarding limited slip differentials, I have spent a few years engineering lubricants for these mechanical systems. One of my formulations served as the lubricant for a famous American two-seater from 2008-2020 when the mid-engine C8 was introduced.

Testing limited slip lubricants, I learned quite a bit about their mechanical design and complexities. To summarize -- absent significant left-to-right or right-to-left bias torque, you will not be pleased. The OEM clutch components and spring forces do not provide that real left-right equal torque that trucks need.

If you upgrade, seek a true "locker" which mechanically eliminates the differential gearset within the rear axle, for a moment or two. Torque will be split equally.

2 Mustangs and an Expedition with mechanical LSDs. They are worlds better than Toyotas electronic LSD.
For my wants/needs better rear end stability in sloppy weather and in rain without having to do anything is far better than having to remember to hit the button in the Toyota then have ridiculous sounds coming from the rear end because it’s trying to do it with the rear brake pads. You can also just really feel it working and engaging.

With a mechanical limited slip it just works when it needs to work. With what I use the truck for a locker is overkill.
I am not rock crawling or getting into stuff where I am really getting stuck in mud.

My 2015 Expedition EL with 3.73s and a mechanical limited slip differential really surprised me at how sure footed it was wet or slippery conditions of daily driving and the mild off-roading that I do.
I’d go to say that if I had it out today instead of the Tundra I would not have needed 4wd. I have pretty much quit even using the eLSD and just put it in 4wd because it is so loud and clunky.
I should have given it a try today to see if I even needed 4wd.