The fuel pump would still run, even when the filter is blocked. Unless the pump motor has packed up completely.
The pump assembly's design is stupid. All these submersible pumps are cooled by the fuel going through them. On this particular design, the fuel flows through the filter after the pump motor, the filtered fuel then passes through the pressure regulator before heading off to the engine. The pump motor is capable of supplying much more fuel than the engine needs. The excess is returned to the tank via the pressure regulator bypass port. This means that the pump/ filter effectively filter the entire tank contents over and over again while the engine is running. This is where things go wrong. Over time, the filter begins to clog. As the filter clogs, the pump motor works harder which increases it's current consumption. This increase in current flow causes the carbon brushes to erode, clogging the filter further. This cycle continues until the pump motor packs up completely (remember it's cooled by fuel flowing through it) or the filter becomes so clogged that not enough fuel finds it's way to the injectors to run the engine.
You can test fuel pump current at the relay. No current would mean the pump motor has already packed up. A high current could mean a clogged filter. A new pump with clean filter should consume around 8 Amps. I've seen a pump with clogged filter consuming almost 15 Amps.
The fuel pressure should be about 65 Psi at the test port, even when the engine is running flat out.