If you have a winding reading through the pump electrical connector, I would believe the pump is OK
You don't really hear the pump, it's the regulator releasing that you hear, on the return line.
You said you had error code from the pump/relay - if you clear, does it come back?
Note that there is no 'sense' for the pump - just the relay.
The relay gets ignition power on the coil + and enabled by the 'low' return at the ECM; when there is no 'low' it will sense the 12V (from ignition) through the relay coil, or post an error if not there; note that it would post the error if you had the relay removed while troubleshooting with ignition on; once posted it will remain there.
So install relay, clear codes and see if/what returns.
If error returns, check the ignition power to the relay coil+ terminal and of course the relay itself.
triumph uses european relays, 2 of the pins are reversed in respect to american relays. you can pull out the spade connectors and switch them to convert to american relays or you can buy HELLA relays.
I would not recommend this - yes, it will 'work' - however the OEM-spec relays are configured with dual-suppression devices to protect the ECM from turn-off spikes; there is a diode in series to the low side of the coil and resistor across the coil itself.
In this case, I would advise go OEM.
One reasonably common problem where appears pump is not running, is that the hose to/from filter ruptures or blows off - this happens if the pump is dead-headed into the sealed connector (remember the regulator is after the return); it may also have happened if your hose clamps were not tight enough when you replaced the pump - but only way to check really, is to pull the pump.
And speaking further of 'earth' (ground)
Does your Sprint have a ground block?
That '99 is unique single-year-model Sprint, which shares the same ECM as the T509 (Speed Triple) and T595 (Daytona), both of which utilize this scheme:
Not sure if your bike shares that architecture ........
Looks like this, where all the ground returns come back into a single connector block
(and the burn is a common failure mode)