I think the answer is yes. The green top together with the roots supplie the nutrient for the bulb growth. So when you cut the top, it will spend all its energy trying to regrow leaves. I do cut the top of leeks and use the greens as chive/scallion. This does not matter, because I don't grow leeks for bulbs.
Even drastic reduction of the number of leaves of onions and galics can adversely affect the bulb size.

Are you growing hardneck or elephant garlic? Those grow scapes which you can cut off and eat. I eat my scapes, and it doesn't seem to hurt the bulb.

I see you live in Austin. Elephant garlic is easy to grow here, but some hardnecks won't do well in our area, and some will. You'll have to experiment. Georgian Fire is doing well for me so far.

If you are going to grow a section of garlic just for green garlic, you could plant smaller cloves for that, and bigger cloves for bulbs. Small cloves don't make nice big bulbs anyway, so might as well.