My experience with Arctic Silver...

If you apply the thermal compound wrong then you can't blame the compound for not being efficient. The application of the thermal compound should be in a very thin layer and that thin layer should be even on all sides. When you apply it it's not a good idea to be in a rush and you should take your time to do it properly so it's done once and you don't have to do it again.
It sounds like you want to have the thermal compound preapplied so you just take the heatsink and place it on the cpu.

AS5 takes 200 hours to burn in.

Did you let it settle in? It takes 200 hours. I got MX4 instead and I recommend it.

Alright, so it just takes some time to settle in? I see. Also, if I got some on my hand by accident and washed my hands a couple times, should that remove it? Also, is it safe or not to shower with the same soap that I used to remove the Arctic Silver from my hand?

Don't worry about the hand or the soap. It's not toxic at all unless you're allergic. If you're still worried some automotive hand cleaner is best for grease.

Yes, AS5 takes a number of power on/off cycles to reach full efficiency. But that is a very minor difference, and I doubt that is the source of your issue.

More likely, the application was not done well.
A common mistake is to apply too much paste. The purpose of the paste is to fill in microscopic air holes in the mating surfaces.
Air is a poor thermal conductor. The as5 paste is also not a great conductor, but is much better than air. If you apply the paste too thickly, it will actually act as an insulator. The AS5 web site only includes instructions for applying it to a cpu.
In that application, only a small drop is needed, and the pressure of the cooler mount will make it spread.

You might reapply the paste, using less.
Clean off the old paste using alcohol, the purer the better.
Use a paper coffee filter for cleaning. It is lint free.
Lint or oils from your fingers can hurt cooling.
It is not harmful to your hands, alcohol works well there too.