The 'short, fat' look occurs if you have the TV set to stretch a 4:3 image to fill the screen horizontally. My flat screens also offer two non-distorting options for 4:3 images. One option is to show the full 4:3 image without stretching it horizontally. The screen is filled top-to-bottom but has blank space on the sides, kind of the reverse of when movies were shown in "letterbox" format on 4:3 CRT screens back in the '90s. The other option is to zoom the image to fill the full screen side-to side and cut off a bit at the top and bottom. I usually do the former. The blank space at the sides doesn't bother me.
Vintage TV sets are cool as historical artifacts but unlike, say, vacuum cleaners, technology has passed them by. To me, an older, even 1990s vintage, TV set just isn't a workable option anymore but I do prefer older household appliances like vacuum cleaners, mixers, stereo equipment, etc., because they function better, are built to last--and more importantly are designed to be repaired--and can be had for a fraction of the price of their modern plasticrap counterparts.
That said, I do think digital broadcast TV is a total crock in that reception seems to be pretty much all-or-nothing. And I really have to laugh at how predictive of modern digital TV stutter Max Headroom was back in the '80s. I'm fortunate where I live that I can pick up about eight stations with an antenna, giving me about 24 channels and sub-channels, but my girlfriend, who lives only four miles away, can pick up only one station. For me, Internet connectivity is key to TV viewing. I stream pretty much everything these days, watching it on MY schedule. Thanks to my ROKU box, 'Nothing on TV' is now a technologically obsolete concept.