Most of the time that is correct. However like I said earlier using my own pool as an example, when my filter gets stopped-up it creates a backlog and water can't flow through, so it churns in the basket (and filter I suppose) and creates a sort of outgassing effect. When my filter gets dirty like that, my pressure goes up, pump basket water level drops, and I see some bubbles from the returns.
In your case however, it doesn't sound like that's an issue since you opened up the filter, inspected, and gave it a cleaning. That sends us back to the suction side. I would try the snake or water pressure "backwards" thing a bit. If you don't have a bladder, get a garden hose and insert it into the pump basket to push water backwards towards the skimmer. Switch teh 3-way valve from time to time. Maybe something is jammed-up in a 90 degree or something just before the 3-way. It's happened before, but usually in areas where they have acorns and such. But you never know. If the lines are 100% clear, no breaks detected, etc, the only thing remaining I can think of is the pump itself.
By the way, who installed your new pump?