Change PCS deployment process to match Build Insights

As a follow-up item from the 1CS review, update how PCS is deployed so that it aligns with the deployment approach used by Build Insights.

## Context
Part of the 1CS follow-up work tracked in the parent epic. See https://github.com/dotnet/arcade-services/issues/6448 for related 1CS workitems.

## Tasks
- [ ] Review Build Insights' current deployment process
- [ ] Identify gaps/differences between PCS's current deployment process and Build Insights'
- [ ] Update PCS deployment pipeline/process to align with Build Insights
- [ ] Validate the new deployment process and document the changes