We aren't a family of 6, but often we will travel with a couple of extra people - friends or grandparents, etc. For shorter trips, there are a few chains that typically have larger family rooms - not really suites, just single rooms that sleep 6. Couple that with free breakfast at all these listed below so I don't have to figure that out for anyone and it's generally a hit.
I'll second Hilton Home 2 Suites, we've stayed in their 2 Queen plus fold-out couch configuration. Embassy Suites also has rooms like that. Marriott Residence Inn and Hilton Homewood Suites also have 6-person rooms for not much more than the standard hotel room. They sometimes have a small kitchenette too for leftovers or making cheap meals in the room. Most of these rooms can be 2 adults and 4 kids, or 4 adults and 2 kids. I'm not sure about 6 adults though, depending on the ages of your crew. On longer trips, we look Airbnb or VRBO for larger groups.