I don't see how tightening screws would do anything to reduce such in-rush current so I would not bother doing that.
That's exactly right, while it may be good to make sure your screws are all tight, in this case you would be INCREASING current delivery. I've seen this happen before ages ago.
The lack of resistance in the 10 gauge wiring is INCREASING the inrush current and dropping the voltage at the panel until that current is reduced. A loose outlet screw (unsafe!) or 12 gauge/14 gauge wiring would all increase the R slightly and REDUCE the inrush current.
Some amplifiers (Luxman for instance) include a slow turn-on circuit, which limits the inrush current during startup, then gets out of the way, but they are uncommon. This circuit introduces significant resistance in the AC until the large power supply filters are charged up and then shorts out of the circuit when the idle current has stabilized.

