Just ordered a PV300 30 amp power supply for my Silent Charger. Ac is 60 watts max, DC is 120. I am going to upgrade to a Hyperion Duo or TME Xtrema soon, but for the time being... Anyhow.... Will the increase in Watts help my 4s 5000 charger quicker?

No.

I'm assuming the supply does DC out at 12v, which would give it a 10 Amp output as well (Power = Current x Voltage).

A charger can use this any amount of this 10A to charge the battery, depending on the charge rate specified.

Judging by your battery, you can charge it at 5A, which means the charger will pull only half (roughly) of what the supply can output.

I may have missed something, someone please correct me if I did.

The only way to charge lipo quicker is to charge are higher than 1C charging rates. For your lipo, the 1C rate is 5amps. If the lipo label says you can charge at 2C, then you could charge at 10amps if you charger would allow it which it unfortunately will not. The charger does not have enough output power - 10amp charge on a 4S would require 168watts and your charger doesn have that much output power even on DC.

Answer is no, simply increasing power supply watts does not increase charger output power. The charger's output watts is set by the manufacturer and the limits of the internal components selected for the charger.