I suppose the mish mash works.  I started without even owning any books with a group that already was playing.  I didn't get my own copies until later.
However, for AD&D, you had to play a mish mash at first.  They didn't have the DMG and such stuff, or at least I didn't have it when the PHB came out, so you used the original rules for stuff that you didn't have yet. 
I actually played AD&D before I ever played Basic (any version) D&D...or BX or BECMI or RC.  (If I remember correctly, it's been a LONG while since I first started).  I think I may have had Holmes at some point before I actually had all of AD&D, so it is possible I may have done Holmes prior to playing AD&D complete????
I think I had one of the earliest versions of Holmes (maybe even a prototype?) as my original copy has something no one else has ever said was contained in their (elves could choose to play as a Fighter or Magic-User, where as the Holmes that I've seen since and from what others say was in their copies, the Elf was the standard Elf which was both a fighting man and Magic-User at the same time).
Never used Holmes in conjunction with AD&D though.