I’ve been lucky and never had to “shop” for a chair since I’ve had access to member sales, etc. (I’m a furniture designer with a large manufacturer), but I’d advise getting a chair with adjustments so you can dial in the fit and comfort. Also, find a place where you can go and test in person. I love classic chairs, but some just aren’t that comfortable. Get one that’s ANSI/BIFMA rated, has a good synchro tilt control (not a swivel tilt or assynch ), 2 or 3 way adjustable arms, seat depth adjustment, and a good warranty.
I’ve sat in MANY of our and competitors’ chairs and for me the most comfortable is HON Nucleus. The seat is extremely comfortable (no seat board so it never will sit hard), ANSI/BIFMA rated, 12 year warranty. Not a classic, but a well designed, clean looking chair (no, I didn’t design it 
) Depot sells these and looks like there’s a model that’s just over $400 new. Sounds like I’m pitching hard, but I know seating fairly well and it’s a great value on a seriously comfortable professional chair.
Allsteel’s Acuity chair is super cool, too, and you might be able to find one used for around your budget.