Maybe some facts about the A320 Fam. and its PTU:
1. There is no selftest of the PTU. The FWC only check that the pressure difference of system G and Y is not more than 650 psi at least 4 seconds after second engine master switch is set to on. (PTU is inhibited with engine master switches in different positions, NLG compressed and either parking brake is on or N/W STRG off)
2. B and Y hydraulic systems have a elec pump. (in normal config system B is only pressurized by elec. pump)
3. PTU connects G and Y hydraulic mechanically.
4. the logic to avoid a overheat situation due to PTU overspeed condition works (simply said) by watching the low pressure status of system G or Y (6 seconds is the trigger). If there is a reservoir low level, the PTU would not be able to build up pressure in that system.
My thoughts to the linked videos:
It does not sound like a hard working PTU so no big problem, but a snag that should be fixed.
Continuous running of the PTU without excessive use of hydraulic (e.g. L/G or flap/slat movement) is not normal. I would first check both EDPs for health.
If one EDP is at the upper end of its range (3200 psi) and the othe one is at its lowest end of range (2800 psi) the PTU can step in.
Other possible triggers for a PTU running during normal operation can be a to high internal leak of a hydraulic system (e.g. partially open check valve) or the breakout pressure of the PTU is to low.