I thought self tacking meant no adjustment of the sheets was required. For that to be so you with a boom you either need a track or presumably a central point, with a single sheet. In your case since you have a track the function of the boom is basically so you can adjust the foot tension.
Otherwise you are dependent as with a normal jib
on the angle of the lead in so far as it divides the tension between leech and foot.
I don't know the boat but if it is designed as a cutter it should go better with both headsails. If it is a sloop with in effect a removable innerstay then that provides a better setting alternative for reduced sail than a partially furled genoa, as does the staysail on a cutter. I suspect in your case it is the former from what you describe.
You can remove the boom but then you are dependent on the angle of the lead through the fairlead on the traveller which can be a limited length to the tack perhaps a few inches thus fixing the angle and thus sail shape. It might be clearer to see as akin to a fixed jib
fairlead rather than one on a track. Further as you come off the wind it seems to me, because the fairlead is further forward and inboard you have less control over the sail shape. You could overcome this if you wished by having barberhauls but I would imagine that is far down the track on a cruising boat.