I have a number of old analogue TV's I use a set top box to convert from digital to analogue I bought the last one about 3 years ago, it was from Maplin make was Icecrypt not sure if
this model but if not very like it.
It was expensive, and the internet bit never did work, but with a hard drive plugged in it allows both freesat and freeview to be watched on any TV connected to it and recorded. The freeview worked well, but since it is continuous channel numbers the freesat ends up with an odd number system and you have to manually arrange the order. The freesat electronic program guide is not very good, it shows the guide for some channels then to get more you have to change channel. Yes you do get all the TV channels but finding out what is on not so good.
It can either combine multiplex with analogue or keep the separate, and I found they interfere with each other so only pipe the analogue signal around the house. The TV with the HDMI connection gets HD from the box, TV's with analogue clearly just get normal quality.
For what it cost I also bought just over a year ago a 42 inch HD Smart TV so not really value for money, may as well buy a new TV, but does do what you ask, it converts all digital into analogue. The record was a surprise, I expected it to change to the channel set to record, however it often seems to record one channel and show another. Maybe they are on same multiplex? Also the format it uses to record, after recording I can select if text is shown or not.
Why one would want to use an old TV I don't know, my tablet will show many TV programs so the TV has never been taken into the caravan. But suppose it would work if I wanted to take it.