P4W: One Board is not enough. We need Multiple Boards and different Task types

Excellent observation about any PPM/Task Management application. It's taken years to realize that my Microsoft Project schedule is built on the foundation of
1. Deliverables - What are we building? 90% of the time I'll create an "Initial" or "Draft" deliverable or a specific name to break up my deliverable to make Progress (%) earlier and identify problems earlier
*Note: Deliverables in the schedules should NOT be tasks lists or punch lists that can be done interchangeably
2. Deliverable Dependencies - If I can't identify dependencies between deliverables due to how the work is done or how the Resources are tied to them, then I'll rethink what they are or if I'm too granular
3 - Resource Modeling - Going along with 2, its important to identify the deliverables in a way that we can model the work of the resources. If you don't resource load your schedules, you're not able to answer the questions of: "Can we do this?", "Is it achievable?"
To answer your overall dilemma, I highly recommend using Microsoft Project Online for your Deliverables, Issues/Risks, Variances. If you have a schedule already, it will import directly into Project Online. And the huge win is that you can surface it in Microsoft Teams to your Team Members to push updates to you versus the traditional, go get progress updates from them.
Then the BIG key and gamechanger is that your schedule uses the foundations of 1-3 above, which is agnostic to phases, methodologies, processes AND the lower level details, punch lists, task lists can be maintained anywhere in ANY app. Using Project Online/Power BI and surfacing within MS Teams gives your Team Members, One-Stop-Shop for the schedule and You and leadership have full visibility of ALL projects, departments and work type.