AppleSoft BASIC is MS. Integer isn't.
True, but AppleSoft BASIC, which came in the ROM of Apple II+ and later, brought floating point libraries and higher resolution graphics.
What do you mean "most of the functionality was provided by it"? DOS obviously refers to the *disk* interaction, and BASIC has none of that.
You assume that all the functionality required by the software at the time was solely provided by the DOS. AppleSoft BASIC being in ROM provided some functions to programs that called them (like the aforementioned FP routines). A practical example being that unlike the Franklin which didn't survive the lawsuit, the Laser was a "clean room" clone of the Apple II+. The feature that made Vtech's Laser compatible was their ability to license AppleSoft BASIC directly from MS. This was important since, despite most commercial programs were not in BASIC, they did take advantage of some functions provided by the ROM including AppleSoft routines.