A permanent relationship is not possible. The same platoon, or same company, cannot continue to be the go-to guys for assault after assault. It's a simple matter of burnout. Higher end folks like the SAs can afford the specialization (e.g. counter-terrorism troops), but when they deploy to a conflict, the personnel start to become fairly "vanilla".

As for the impact on the ability to conduct constabularly or defensive missions, there probably is none. Just look at the multitude of MOS's that are conducting SASO right now. It's a just range of other skills.