There is such a thing as the "social contract" and traditionally it has meant showing consideration for others through a neat and appropriate appearance, among other things. Garments worn for sleeping aren't generally considered appropriate for dining with strangers at a public restaurant.
Yes, but unless it is a posh/exclusive restaurant that has clearly-publicised 'standards' or rules, different people care vastly differently about how others dress in public.
Near where I live, there's an older lady who is never seen in public in anything other than full Victorian formal dress - like the people you used to see on Quality Street tins. She does look amazing but, for all I know, she quite possibly considers everybody else in the shops/high street who are wearing common everyday modern outfits as the height of slovenliness and 'too lazy to make an effort'. Then again, it's just as possible that she has her own style that she personally adores and fully accepts that other people are not her and they 'do them' just as she 'does her'.
As I said upthread, most people in a relaxed/informal public place really aren't that bothered by what strangers wear or do.
Would you like it if your wedding and funeral attendees showed up in PJs or speedos? If a parent greeted you and your offspring for a playdate wearing a G-string and mesh t-shirt? If your teen's prom date showed up in their underwear and a robe? After all, comfort is king, eh?
I already clearly pointed out the marked difference when somebody is being paid to do a job where they are serving you/others, as opposed to being at leisure.
Yes, a g-string and mesh t-shirt are extreme clothing choices for in public. A blouse/shirt and trousers/shorts that happen to have been designed for comfort and are thus commonly worn as nightwear are not.
When you say 'underwear and a robe', you do realise that a robe is basically a garment that covers the whole body and often used as a synonym for a dress or gown? Are you really asking if I would be shocked if a teenage (presumably) girl turned up for a date with my teenager wearing a wrap/other style of dress and (potentially/hopefully) unseen knickers underneath?!