The reason why you can't move people on and off scholarship in "head count" sports is that it would be abused. "Head count" sports and "equivalency" sports are completely different animals.
With the "head count" sports the rules are extremely complex with regards to "counters", "initial counters", "participants", what constitutes a "recruited athlete", etc. The reason the rules are complex is that the NCAA wants to make it virtually impossible to find loop holes to circumvent scholarship limits. Even with these steps, you still saw SEC schools oversign and use blue-grey-purple-brown-fuscia-whatever "shirts" to try to gain a competitive advantage.
Oversimplifying things for brevity's sake... if a "recruited athlete" that was an "initial counter" is receiving scholarship money in any way they are still a "counter" towards whatever the "head count" limit is. Otherwise, teams would find a way to play the shell game with "participants" versus "non-participants" to keep the guys they know aren't going to play uncounted in a given year.
If a student-athlete is removed from the team, enrolled in school, and not on scholarship they generally are not a counter but there are also lots of times where technicalities would make them a counter depending on timing and eligibility. It's really case-by-case... but yes, you can be off scholarship and still count towards the "head count" for the year if certain criteria are met.
Finally, generally you can receive scholarship and not be a counter if and only if you sign something forfeiting your eligibility (e.g. a medical, etc.). If you have eligibility remaining that would allow you to be a "participant" in the future then you generally can't be taking any money that isn't strictly need based without counting.
