286 protected mode (PM) is fundamentally different to what the 386 offers. Think of the 286 PM as a prototype, which had so many shortcomings that almost nobody ever used it, and the whole thing was completely redesigned from the ground up for the 386.
It did not use a flat memory model, it used a segmented model like real mode, which meant you had to jump through hoops to access memory in blocks larger than 64kB at a time.
It was completely incompatible with all the (MS-DOS) programs available at the time, so once you were in PM you couldn't use any of the programs you were used to.
You also could not leave protected mode again unless you rebooted the PC, so manufacturers came up with creative solutions like putting a flag in RAM, then writing a magic value to the keyboard controller, which would flick the reset pin on the CPU to reboot the machine. The first thing the BIOS would do is detect the flag set earlier, where it would then jump back to the original program instead of running the POST routine, allowing the original program to continue running having "exited" PM.
This meant that using the 286 PM prevented you from running normal DOS programs without a large number of tricks. At a time when there only were DOS programs, it was not worth the effort of using PM at all.
So it ended up being more complicated to work with the 286 PM than just living without it, and relying on EMS and XMS to access the extra memory. A number of 286 motherboards had chipset support for EMS so that you could use all the extra system memory without the need for PM.
Intel recognised these shortcomings and produced a brand new completely different PM in the 386. The flat memory model makes accessing memory in a chunk of up to 4GB simple. The CPU can go in and out of PM with a couple of instructions so no clumsy reboot protocols are needed. VM86 means that most of the time you don't even need to exit PM, you can run DOS programs while still in PM.
All these improvements meant that the 386 PM was not only more functional, but significantly more compatible as well.
In other words, the only thing in common between the 286 and 386 protected mode is the name. This is why PM operating systems are typically 386 or newer. Adding support for the 286 PM would be an entirely independent effort, with little or no code able to be shared with the completely different 386 PM.
By contrast, the 386 PM works in much the same way right up until the last of the 32-bit CPUs, and even beyond if you run 32-bit software on 64-bit CPUs.