I personally can't see any particular reason why a lab type PSU can't be used. As long as the power requirements are met any PSUs should be used. Most lab type PSUs have really good dynamic response, so shouldn't be any problem.
Perhaps they have tried with a lab type PSU having a current limiting (CC) feature and experienced some problems. As you might already know, those PSU decrease the output voltage if the current limit is reached. Most of the power is drawn by the backlight. At startup the backlight driver may draw excessive currents which may trigger the PSU's CC and therefore decrease the output voltage to a level that is not sufficient for the rest of the circuit to operate.
Or, they don't know what they are saying.
You may want to contact the seller again and ask the reason behind it.
IMPORTANT NOTE: The backlight driver in the 1st picture (the board in the middle) is specifically for CCFL (cold cathode fluorescent lamp) backlights. You can't use it with LED backlights. The mainboard may have a backlight driver for LEDs. So, be careful before using.