Does holding a Temporary Resident Permit fix my inadmissibility for future trips to Canada?

No, not on its own. A TRP authorizes a specific stay for a specific reason despite an existing inadmissibility — it doesn't erase or resolve that inadmissibility, and it generally doesn't automatically cover future trips beyond what it was issued for. Each new trip, or each new period of authorized stay, generally needs its own consideration of whether the person is still inadmissible and, if so, whether a new permit or another route is needed.

There is a narrower situation where holding a valid, unexpired multiple-entry TRP can support more than one trip within its specified validity, but that's different from the underlying inadmissibility being fixed going forward — it's still the same permit covering the same authorized purpose and timeframe, not a permanent resolution. Rehabilitation, whether deemed or by application, is the route that actually resolves the underlying inadmissibility for the future rather than working around it trip by trip.

Anyone who expects to travel to Canada repeatedly, rather than for a single occasion, should think seriously about pursuing rehabilitation alongside or instead of repeat TRP applications, since relying on permits indefinitely tends to be less certain and more work over time than resolving the inadmissibility itself.