Unfortunately, not only are many converters not built to the same standard, not all devices like them. It was a bit of a struggle to convince my new TV to run my old RF connected units.
If your family members can't help, I'd hit up some local thrift stores- finding a small TV new to be a second unit is very difficult these days. 40-42 inches has become the standard low end. A thrift store is going to be more likely to have an older, smaller set around. Alternatively, they tend to have a lot of LCD monitors, which could be a good option to get a smaller HDMI capable set as well.
If you do decide to buy a new set, visit
rtings.com to do some research. They have a really robust testing setup that goes over a huge amount of data. I found them looking for a place that could tell me a TV's input lag (very important if you're hooking up retro to a modern set.) Plus, they do not accept review TVs from companies- everything they test, they go to a store & buy themselves. That way, the companies can't send carefully chosen, better preforming TVs, & they aren't biased by who's giving up free stuff.