As a long time owner, and talking to lots of tesla people, the charge cables can sometimes be on the wrong side of a tolerance stackup and be stiff. The easiest method, on the cars that were the hardest to plug in, using UMC, HPC and Superchargers were to align the plug, and then just firmly shove it in. We joke about using your hips, but it is actually easier that way. Model X and 3 might be different due to the height of the chargeport. Just hold the cable with both hands, tuck your elbows into your body, and then use your body weight to firmly insert the connector. Don't worry, you wont hurt it.
The supercharger does monitor the temperature of the charge handle as it charges. If you feel it get hot, then it might be limiting charge to account for that. This is common as any added resistance to the connector creates heat. And that resistance can change from different insertions. So, one quick test is to stop charging an replug it in. If nothing changes, then the contacts are probably at their best.
I have used charge connectors that were dropped by other users, or have sat in mud etc. Not knowing this, I have . felt that super gritty connection when plugging in. I have yet to have any problems supercharging.
Also, the Supercharging cabinets are made up of lots of the same chargers that are in Model S/X. This scale has helped keep the superchargers costs down, but has created the same redundancy that the car battery has. If a cell fails, the rest of the cells in parallel can still operate and keep working. If a charger in the supercharger fails, the suoercharger will still work with a lower output. This is also another reason why a charger might not put out full current.
