The US regimental system didn't really work along the same lines in terms of budget (although the original 2nd Cavalry was an exception...it was Jefferson Davis' pet regiment when he was Secretary of War) or perks. You did see squabbling from time to time between the combat branches, but most acrimony was reserved for Staff officers.

In the Old Army most resistance to change surfaced with senior branch officers and not within the regiments per se. There was some fussing and resistance about mechanization among cavalry units in the 1920s and 1930s, but part of that had to do with the fear that a new armor branch would cut into their traditional roles (which it of course ended up doing...but then armor became as protective of its roles and uses...).

The biggest issue with the old regimental system in the US had to do with how promotions were handled. At the time it was by regimental seniority first, then by branch. That was later changed, but in the years after the Civil War there was a tremendous amount of rank stagnation.

There are certainly regimental negatives, but I'm not sure they outweigh the positives IF the system is designed correctly.