It is very unlikely that the queen signs anniversary cards personally. I have some experience of receiving post from the queen. I sent her a letter about a charity I am involved with. I got a reply but it was signed by her Royal team, not by the queen herself.
She hasn't got the time
Think of all the post that must be sent out in the queen's name every day. All those 100th birthdays, all those formal engagements, all those charitable requests, all those letter bestowing Royal status on jams and stables and retailers... It would take far, far too long.
She has better things to do
Walking the corgies, polishing her crown, waving at people, smashing wine against boats - the queen likes to keep busy but not by repeatedly writing her name on bits of paper. She's far too expensive for that.