My Controller Only Charges If I Hold The Cable At A Certain Angle

You are right about the cause and your test result confirms it neatly. Three cables behaving identically, and charge state flickering as you move the connector, means the fault is in the controller's socket rather than in anything you plug into it.

Specifically, this is cracked solder joints where the port's mounting legs meet the board. Micro USB connectors are mounted with small surface pads plus a couple of through-board legs, and every time the cable is flexed or levered the joint takes the strain. After a few hundred cycles the joints develop hairline cracks. Holding the cable at an angle closes the crack enough to conduct, which is exactly the trick you have discovered, and knocking the desk reopens it. It is the single most common failure on a DualShock 4 of this age and the reason the DualSense moved to a more robustly mounted USB-C port.

On safety, since you asked, and this matters: it is not dangerous in the sense of a fire or shock risk. A cracked joint makes an intermittent connection at 5 volts and low current, and when it breaks, charging simply stops. What it does risk is the battery and the board. Repeated interrupted charge cycles are hard on a lithium cell, and if a joint tears completely it can lift a pad off the board, which turns a cheap port replacement into a much harder repair with a wire bodge. So the honest advice is that there is no urgency today but do not leave it for six months.

On whether to repair: yes, if you are comfortable with or can pay for soldering. A replacement micro USB port is a very cheap part. It needs a soldering iron and a steady hand, because there are small pads close together, but it is a routine job for any phone repair shop and should be inexpensive. If the controller is otherwise healthy, that is clearly better value than replacing it.