They do have in mind the perpetuation of their own brand of religion/culture/civilization, i.e. of themselves.

Yep, that's how it works. ;-)

They want to see that there are more of their kind on Earth so they'll inherit it (though this doesn't sound meek to me). The same guys probably aren't happy with their competition "breeding like rabbits".

Perhaps there is another explanation. Perhaps grandparents love bouncing grandkids on their knees and enjoying the thought that some part of their own life work, values, whatever will outlast them. It's not satisfaction that they've bred more effectively than some sort of enemy, it's satisfaction in family and relationships for their own sake. It's a good thing and it isn't proper to recast it with Nazi motives like that.