Illegals were/are special assets that would never be concurrently running so many different operations and engaging in risky wet (i.e. assassination) operations right and left. More likely they would spend many boring years cultivating their positions in society and a select few important contacts. That doesn't make for good television.
The Sopranos was not especially realistic. But realism wasn't the point; there was just enough verisimilitude to serve the narrative, which was a kind of morality play. It's the same with The Americans, which is at bottom more of a relationship story than one about espionage. (We agree, I think.)
A lot of research went into the show, but it shows up in the same ways research shows up in Mad Men.
Don't watch it expecting to learn a bunch of stuff! That's not the point.
(A top 5 series for me.)
As drama, it excels in the drama around the marriage rather than the actual fact of them being spies, and has been praised as “fundamentally a show about a marriage”. If you’re looking for a spy thriller, you might look elsewhere. It’s very “American TV” and doesn’t really stray from the formula