I am not sure but I think those who use Aluminum Jelly use it after most of the dirt has been removed and it is only used to bring out the Gilt left on a button. Not sure, but I am fairly certain most still clean by various methods to removed the stubborn dirt first. I know since I do not have Aluminum Jelly and do not find enough buttons with gilt left to justify buying it and keeping it on hand, I use a lemon for the final bringing out of the gilt color and it work just fine as long as you do not leave it on very long at all.
I have cleaned all my buttons using the peroxide method and have no qualms, yes to get the backmark readable sometimes requires more soaking and cotton swab cleaning but it still works without changing patina color.
I have seen an old post from 1999(TNet) where someone and I think I know the fellow used Top Job soaking to remove the dirt for brass buttons only, NOT PEWTER.

I have never tried a cleaner like that and would not care for the odor to be in my working room so I will stick with peroxide.
For those who soak in olive oil, well once you do that and it does not work great, you are basically stuck with what the result is, the oil is very hard to remove afterwards and never do it on two piece buttons, the oil soaks inside the two halves and leaches out for years afterwards, been there, done that and no more.
Don