If I may chime in.
When I was growing up, we were told somthing that sounds very simliar and it could well be that it’s true and you don’t get the concept. And by the way becuase I was brought up this way I can’t stand when my son dose it so I will ask him to wear slippers. Some children & even grown ups have a habbit of kicking there shoes off under the table. We were told that it’s not Shabbosdig to go around with out shoes because an Avul go’s with out shoes during the week. That is still true today, So basicly you can understand the concept. Now for you to get the kids to not kick there shoes off is a chalange and for her to get that is even harder because she had kids a long time ago. And I think you can tell her I understand about the not wearing shoes on shobbos, but for kids its hard and you will work on it at home.
About her having different opinons is like this. If it’s in her house she can have her rules but if they affect you? what ever they are you need to decide if you can live with them or not. If not, then try not to go there. Trust me, It took me a while. I got very nervous every time my father inlaw bud in everytime I had somthing to tell my young children. Every time I sang a song he told me how it go’s different, and some other things. I didn’t argue with him. I just stopped going there for Shabbos at one point. I have all different excuses, not to go there and they are all true. That’s what he accomplished. My biggest excuse is the best. It’s Shabbos and I need my own bed. I’m a little older than I was ten years a go.
In your house? you are the boss. Yes, every one gets respect, but the rules are yours. If you say plastic plates then its plastic. no arguments needed. I think the problem is, and corect me if I’m wrong. You start feeling put down, if everything is not the way she wants. But you know what it dosnt matter a bit. What you could do is, next time she comes serve everyone on plastic and serve her on china if she asks you why say respectfuly. We usally use plastic to make it easier for us on motzie Shabbos but I know you like China so L’Chovod You I took out china. If she has brains sh’ll say you dont have to.
But then on the other hand. It could well be that your MIL was brought up in a home where they stressed issues on Chovod Shabbos, that’s what it sounds like. About the shoes, and here about China. My grandparents hardly took out a plastic plate and buying a dishwasher was never a thought. So that gos back to the your house her house rules.
Is she from Eroupe by any chance? over there they are very in to China 🙂