He did! Mary, in particular was betrothed quite a few times as a child but then Henry VIII made her illegitimate and then later her half-sister, Elizabeth, it was not then so important to marry them off well. They were bastards and were useless because other royal families did not want to be connected to illegitimate stock.
As for Henry VIII excluding the heirs of Margaret Tudor from his will, I believe that this was down to them being heirs of the Scottish crown. Henry did not want a King of Scotland on the throne of England! I don't think that Elizabeth I had any choice in 1603, she could have passed the throne to Catherine Grey's son, Edward Seymour, but her council definitely favoured James VI.