Well, turns out I got some good advice from an expert in the field. The USB adapter with the FTDI chip did indeed work like charm. And in Windows 10, no less.
It’s not that it has an FTDI chip, it’s because it supports TX/RX plus the other handshake signals of DSR/DTR/RTS/CTS and perhaps even DCD and RI all at the proper RS-232 required voltage levels.
If you’d tell us the brand and model of converter, we’ll know for the next time someone asks “which USB to RS-232 converter supports the old PC standard pin-outs and levels?”.
I know from personal experience the Digi Edgeport brand of converters are 100% reliable for all applications but most wouldn’t want to pay for them as they run about $100 US.