Would it make sense to try to set my exposure using mainly the shutter speed and the built in ND filters in my XH-A1? The reason I'm asking is because I usually want a shallower depth of field so I need to open up my iris as much as possible. Would adjusting the shutter speed make a difference other than darken or lighten the image?

Using the shutter to reduce the light gives a strobby effect like that used on "Saving Private Ryan". Depending on the action, you can often get away with increasing the shutter speed so that you can open up by one stop (eg 1/50 to 1/100), but it does become increasingly noticeable beyond that.

It's better to use ND filters in front of the lens.

Brian is absolutely right. I HAVE used up to 1/125th on some subjects that have had motion and gone as high as 1/250th for a talking head with little or no motion and "gotten away with it". Use all the other tools at your disposal first and THEN fine tune with shutter if you still "need" to.