Do soldiers pay any attention to the army's regimental system, which is supposed to giver every soldier a "home" regiment and point of pride?

Combat arms still get their regimental affiliation with their first unit and for many that is a source of pride.

When I talk about "MY" Regiments it means the 325AIR and the 75 RGR, I follow them still today, wear the DUI's with pride.

I can say it is something that Battalion commanders don't really deal with. There are very few regimental commands in the Army. For example, 3BCT, 82ND, is not the 505th PIR. Since that is the case the BDE commander can't induct anyone as a distinguished member of the regiment. Since there is not a regimental commander it would fall to the 1st Battalion commander. He is in theory the bearer of the regimental colors.