DOSShell was indeed introduced in MS-DOS 5; for MS-DOS 6, you need to get the MS Supplemental disk; it's also included in OLDDOS.EXE, a supplemental archive of utilities intended for Windows 95.
In any case, it is not really the optimal choice, as it takes up a good chunk of conventional memory and is probably not as well-behaved as the OP would like.
I don't know about previous versions of DOSSHELL but V6.22, like most launchers/shells/menu programs of that era only left a small stub of itself in memory when launching external programs.
Before launching DOSSHELL in MS-DOS 6.22:
After launching DOSSHELL in MS-DOS 6.22:
I haven't tested it extensively, only with a few apps & games and had no problems.
I was just going to say, a lot of dos games recommend against using it in their readme files.
Many games at that time recommended against using ANY shell/launcher, not just DOSSHELL as every kilobyte of base memory was precious.
In any case there are definitely better choices for DOS shells/launchers than DOSSHELL. I merely pointed it out because I was under the impression that it was installed by default. Come to think of it, I have never installed DOS 6 and higher from diskettes. I had a boot diskette where I:
FORMAT C: /s
copy DOS directory from ZIP drive to hard disk
create CONFIG.SYS / AUTOEXEC.BAT
The DOS directory was copied from someone else's PC that had installed it from the setup program.