The 4:00 opening limits your options a bit - I'm sure you realize that though.
I'll throw out a third option: Davanti Enoteca.
Avec is the only of these options that meets all your criteria...it's the obvious choice, but if communal & uncomfortable seating bothers you look elsewhere.
Publican's wine list (especially by the glass) may not meet your definition of affordable and IMHO the atmosphere is more cafeteria than wine bar.
Davanti Enoteca may be located a bit west and/or south of what you are looking for, but does have cheese and affordable wine in, what I would consider, the most pleasant environment of the three.
Davanti Enoteca
1359 W Taylor St, Chicago, IL