I have a lockdown baby. Ive also had a covid infected baby when she was 6 weeks old.
It floored me and DH, she had a cough for a few days and was fine. I would probably be more concerned being new parents of your health as having a newborn when you have covid is very, very hard.
I probably wouldn't ask people to do so, because they're so inaccurate. My parents have my 2 brothers at home and one of their girlfriends. All of them caught covid and only 2 showed up on a lateral flow. All positive PCRs.
Colds, coughs, RSV, any viruses / infections are horrible and covid is just in the mix of them all.
If lateral flows were reliable I would maybe ask while the baby is a few weeks old. A lot of it is common sense. If you feel poorly or think you're coming down with something, don't come!
But with lateral flows being so inaccurate, and anyone can be on the brink of a cold, cough, flu etc. I wouldn't ask. I wouldn't object if someone asked me to do one to see their new baby though, if I was asked to do one, I of course would.
Covid is everywhere and not going anywhere for a while. Someone doesn't have to hold your baby to pass it over. So you can catch in shops, cafes etc. The risk is smaller, but you still can.
We all caught covid from DH's step mum, she didn't make any physical contact with any of us and didn't hold the baby. Maybe if she did a lateral flow before it may have showed up. But from personal experience they're too flawed to take much notice of.