OP, it's clear you've had a lot of rejection in your life and it has hurt you a lot. It is also clear that you don't understand why, and you've said you would like to understand why. People are replying trying to (nicely) explain why and you are responding by telling them they are wrong. You seem like a nice person at heart, and someone who is hurting, so I'm also going to try to explain why - though I suspect you will reject my explanation just like you have rejected everyone else's.
On this thread you have exhibited quite a few characteristics which make it difficult for other people to maintain enjoyable relationships for them.
People have tried explaining to you that different people do cousin relationships differently and that's perfectly fine. You won't accept that. If your SIL doesn't make the same decisions you would then you label her as unkind or not caring. I'm sure you don't mean it that way but that's incredibly controlling.
You say you don't convey that to your SIL but your replies on herd indicate you have little self-awareness about how your own feelings may be conveyed snd how intense you are.
At 7 years old, unless the two cousins are joined at the hip and absolutely best friends (which is clearly not the case here) there is no problem letting a child pick to just invite their friends to a party. Would you do it? No. Does that make it wrong? Also no.
The fact that the child has been told to lie about it is unusual and suggests that the family were scared of the reaction from someone if they found out. From your posts, I think it's likely that person is you. I know you don't think they see/experience a reaction from you, but I think it's highly likely that they do but that you have no self-awareness about it.
You did nothing wrong per se in offering to take the cousin on holiday or for them to come and stay. But your reaction to them saying no shows no understanding of their position - again unless they agree to have the kind of relationship you want then they are seen by you as horrible people. There is no way in the world I would send a 7 year old to stay with or on holiday with someone they do not see extremely regularly and feel very comfortable with. That's not you OP. Making the offer is fine. The level of judgement and hurt feelings when that offer is declined, is not fine.
Your description of the playdates that have fizzled out, also show these patterns of thinking - that if someone doesn't do a relationship the way you would do it that they are bad/unkind/uncaring people (which is not true) and you are disproportionately hurt by it.
Your replies on here are incredibly intense yet don't take on board anyone's suggestions or feedback. If friends/relatives don't follow the same script you would use then its clear very hurt and see them as mean people, time after time.
The other people aren't doing anything wrong per se. They just aren't living their lives the exact way you want them to. The only way that this situation is going to change and be less hurtful to you is for you to develop an understanding that them not following your script doesn't mean they don't care or that they are horrible people. It's clear that these types of thoughts are linked to issues/rejection from your own past and that these are fueling anxieties about your own DC being hurt and rejected. The problem is that your intensity of feelings and rigidity of thought makes this more likely, as it is more likely both that people will avoid you (and in turn your DC) and that your feelings of rejection when people live their own lives, will seep out and result in your DC also feeling rejected when there is no need.
As someone has said up-thread please get some counselling/therapy. I know you see that the problem is "them" (your SIL, the friends, etc) but what is actually causing pain for you - and has the potential to cause pain for your DC - is extreme emotional response to other people's relatively normal decisions.
And FWIW, I never invited a cousin to a birthday party, never went to a cousin's birthday party, never stayed overnight at a cousin's house, never had a cousin stay overnight at my house, never went on holiday with cousins, saw them 3 or 4 times a year, and as adults we are quite close (though still do none of those things!)