Quite sad how many people are defending MIL or suggesting work-arounds or just putting up with it. I mean, WTF?
Assuming OPs family have been turning up in good time, OP needs to be pointing out to MIL that it's bizarre to invite someone to eat with them and then not wait until they arrive to eat with them.
Christmas dinner situation was particularly rude. From the way the tale is told it doesn't sound like OP was meant to be there for one o'clock or something and didn't get there until 5pm? Come on -to know someone is on their way and will be with you within half an hour, but eat Christmas dinner anyway is definitely MIL making a point. If OP was very late fair enough, but if not, then that's just rude. I guess some people are scared of confrontation but this is definitely one of those situations that warrant it. There are lots of people in the world who genuinely don't understand how rude some of their actions can appear. The OP could be doing MIL a favour by pointing it out.
As usual the partner seems to be absent in calling out their own parents rude behavior. Isn't that always the way in these stories, but if I was the OP I'd be quite blunt to MIL and state there's not much point accepting invitations if they have no interest in sitting and eating and chatting with OP's family. No doubt the MIL will have a list of excuses to justify their behaviour and try to make the OP seam like the one in the wrong. Don't have any of that nonsense. Best to have some comebacks ready for the likely excuses.
If the table is too busy with everyone around it why does MIL invite them to dinner (I mean isn't that the point at Christmas anyway?) Same if the issue is the kids behavior, or being private about eating etc. Why invite them to eat if that's not actually what they want/have the furniture for?! I would ask that directly.
If they got hungry waiting, then the polite thing is to do what everyone else does when waiting for dinner party guests to arrive and nibble a couple of crackers/quality streets or something to tide them over.
Alternatively, if they HAVE to eat at a certain time then just agree on an arrival time well in advance of dinner time and then it's on MIL to stick to that set dinner time.
I hope OP discusses it with them directly because on some level MIL obviously thinks this is completely okay behaviour and it's not.