Wait, huh? Let's start from No.
As a practicing synthetic organic chemist, I agree with the statement that silica gel dehydrates solvents by water absorption. Sure. But I've heated plenty of alcohols in the presence of silica gel, and 99% of them don't spontaneously lose water! (That would be a rocking olefin synthesis, if it worked...)
|Methylamine: Easy as hooking up five pressurized reactors in your kitchen sink.|
In this scenario, both are unlikely, especially at room temperature and pressure.
Now let's talk practicality: which company will sell you a cylinder of ammonia gas for 'home use?' (Not Home Depot). How will you get your methanol? What's the plan to isolate the (volatile, stinky) methylamine from the mixture of compounds this theoretical reaction produces?
OK, how do companies make methylamine? Albemarle technical documents to the rescue! Seems that mixing methanol and excess ammonia at 300-500 degrees Celsius, under pressure, over a zeolite catalyst will produce an equilibrium mixture of methylamine, dimethylamine, and trimethylamine (favored). After fractional distillation, the trimethylamine can be streamed over an amorphous silica / alumina catalyst to disproportionate it back into methylamine.
Not a kitchen sink in sight.
Curious - Appended at the bottom of the essay is a thanks for Prof. Adam Braunschweig, faculty at NYU. To what extent did Prof. Braunschweig proofread this post? Did he sign off on the "kitchen sink silica gel" concept in the middle? I can't possibly imagine that he thoroughly vetted this essay.