Although it is a little odd and I'm not certain I've seen it in writing, I have definitely heard constructions like "John will both try and kill mosquitos." to mean, "John will both attempt to and succeed in kill[ing] mosquitos."

"John will both try and like sushi" makes perfect sense, although there's an implied "to eat" verb separate from the "to like" verb in there that isn't present in the constructions the article is talking about.

Likewise, "I tried and finished the assignment," means "I tried (to do) the assignment and I finished it." Again, maybe not in writing, but with a certain inflection on 'tried' (where in writing maybe you'd put a comma or semi-colon to indicate a pause) this is something people actually say; although they may emphasis it with "I finally tried and actually finished the assignment." (Whereas maybe previously they weren't confident they could even do it and maybe didn't try.)

Included for no real reason: "They tried and failed, all of them?" "Oh, no." She shook her head. "They tried and died."