The thing with social mobility is that while you can go up, you can also go down. So insecure people might fear that for them, social mobility would mean to go down. And the higher you are, the more secure you need to be on your work and skills.
On some level, I think that many people who is against social mobility think that they don't really deserve what they have. Yet at the same time, they feel they are entitled to lot of things under the current system with almost no social mobility.
And some of those things they feel entitled, doesn't cost money. For example: having a homemade meal by your wife. Not having to clean the toilet. Get recognition for doing housechores, etc.
They can't get a piece of land, but they think they can get a wife who would take care of them. That is the thing, there is a reward for them under the current system. A guaranteed consolation price which is more than what they might get with social mobility, where if being judged fairly, they might end with even less than they have now.