The programs that you already had and needed to run were not protected mode programs, they needed to be run in 16-bit real mode anyway.
DOS drivers used BIOS for I/O, which also was 16-bit real mode API.
If you simply change DOS to be 32-bit protected mode OS, you need to provide a 16-bit real mode for programs. And for running BIOS code. A lot of switching back and forth between protected and real modes.
The point is if you already have a BIOS interface to access say a SCSI hard drive, through BIOS ROM onboard the card, then you need a protected mode driver for that. Or just run in 16-bit real mode and use the disk adapter BIOS for access.
Many programs bypassed DOS and used the BIOS directly, for various things, be it a video mode change or something similar. So you would need to have an interface anyway to provide access to 16-bit real mode BIOS, and in the case of SCSI disk, if you don't use the original BIOS interface, your driver needs to then provide emulation for the disk BIOS.
If you run EMM386 or similar memory manager, then actually EMM386 has already set the CPU into protected mode for memory mapping purposes, and it has also created a virtual 86 mode task where the DOS then continues to run like in real mode and so DOS, BIOS access and programs can continue to run in real mode. The EMM386 mainly handles the memory mapping to provide rest of the memory as EMM to the virtual 86 mode task.
If you don't use EMM386 or similar manager, but only use HIMEM or similar memory manager to provide XMS, it will also switch the CPU to protected mode momentarily to allow for copying conventional memory to XMS or back, but the CPU will be running in actual real mode, not in virtual 86 mode.
If a program such as a game wanted to run in protected mode, it can still be run in protected mode, and use DOS and BIOS through real mode callbacks of the protected mode extender library the game uses.
As the system worked, there was no need to write a DOS which would run in 32-bit protected mode.
Additional advantage of running in virtual 86 mode is that it enables hardware emulation with a TSR. If you had a Gravis UltraSound, you could simply install a TSR which asked the EMM386 to trap IO port accessess and call the emulator when these ports were accessed. It made emulation of Sound Blaster PCM DAC, Adlib FM synthesis and Roland MPU MIDI interface possible.
So if you wanted to run any program under virtual 86 mode, and under protected mode DOS, you would need to also emulates BIOS interface. And yet allow programs to use the hardware like display adapter or sound card as they wished which bypasses protection, unless you provided an API for that which programs must use - which starts to sound like Windows already.
Additionally, if a program wanted to set up the protected mode itself, it can't be run if the CPU is already in protected mode and the program is started in the virtual 86 mode task, so it would not be compatible. Some demos and games required this and the PC needed to be started without EMM386 driver for them to work.
So a protected mode DOS would not have been a real DOS, it would have been a new OS, requiring the write or porting of new programs, or requiring the write of DOS compatibility emulation layer for running existing DOS programs, and it would have had many places for problems and incompatibilities running old DOS programs.