Well, they aren't really getting than many extra ASIs. They get ASI #2 one level early (7 instead of 8), ASI #4 three levels early (13 instead of 16), and a bonus ASI #6 at level 18.
Putting ASIs at character level makes some levels really dull (nothing is gained at 6 AND 7, for example), and some levels overstuffed (at character level 8, you'd get both 5th level features AND an ASI). I'd rather do something like make the ASI for your second class not count, or only give a +1 bonus for it, or something like that.