Street traffic is allowed through Pike Place Market because the vendors want it that way. The Market is an irreplaceable treasure and I think we generally ought to support the preferences of the people who make it work, for fear of "fixing what ain't broke".
Do the vendors specifically want cars as a whole, or specifically their cars and delivery vehicles? We could allow just the latter, and prohibit cars that aren't in service of the market. Similar to the "Healthy Streets." The rules won't be 100% honored of course, but it'd likely reduce a substantial amount of the traffic there. Especially if towing is on the table for non-Market-affiliated cars parked there.
The vendors want cars in that street because otherwise pedestrians would walk down it instead of on the narrow sidewalk which goes right by their businesses. They literally want the traffic so that people are more likely to go into their shops.
Probably cars as a whole. Based on other such places (I know nothing about Seattle) they have no clue what cars are costing, but they know some people get there by car so they are scared to lose them.
The parking spots at Pike Place dwarf in comparison to the crowds the market serves. Here's a maps screenshot[1] where I've highlighted the parking available there[2]. Meanwhile the market serves ~10m annual pre-pandemic visitors[3], which averages to 27k daily visitors. The amount of visitors coming in by car are negligible.
How much damage can drivers do at 1 MPH, anyway?