IIRC, the US Army abandoned the RCTs in the late 1950s when the 'Pentomic Division' TOE was adopted- it was felt that attaching tank battalions to infantry regiments or vice-versa was too awkward when it came to properly coordinating & utilizing the combined arms in a mechanized division, as well as increasing vulnerability to a tac-nuke, so the regiment was abolished as a tactical unit in the reorganization, being split up into battalions, & the old regimental numbers being retained as identifiers that would preserve something of a unit's history & lineage, though the active battalions of a regiment didn't have to be in the same brigade, or for that matter, in the same division, even if some effort was made to maintain historic divisional affiliations for a few years for purposes of tradition.
In it's place, a division would be comprised of several brigade battlegroups that contained both infantry & armor elements, and supposedly capable of a high-degree of independent action as well as dispersing combat capability in a way that would minimize the consequences of a nuke vaporizing part of the line, modeled on the combat commands of WW2 armored divisions & the German 'Kampfgruppe' concept from the same war. Artillery, air-defense, and assorted supporting assets would be similarly broken up into several dispersed groupings to reduce the risk of a nuclear attack being used for rear-area interdiction crippling a division by destroying its tail.
Although the Pentomic Division proved to be a bad idea that was abandoned after a few years, the regiments no longer existed as any sort of coherent unit, and the brigades were found to be a better way of organizing divisions for combined-arms mechanized warfare.