I haven’t used Moka Pot so I can’t comment on the direct comparison of the two. AeroPress was a little too clean for my taste. Turns out I like a hint of grit in my coffee and and miss it when it’s not there. I got a 12 oz french press, probably knockoff of this model www.amazon.com/…/B00005LM0R and never looked back.

Did you try a metal filter in the Aeropress?
It let’s just a bit of the fines through and would likely provide what you are looking for.

But, that french press looks great also.