I also think that gifts should be given with no strings. The difference between our points of view, I think, is that I also think that for someone outside of the giving/receiving dynamic to dictate the way in which the gift should be given is out of order, and I think that not allowing a grandmother to give a gift that is recognised as being from her is unacceptable behaviour.
If the mother wants the children to think that Santa brought all the presents that the parents bought, that is up to her. To insist that everyone else goes along with that as regards their own presents seems to me unreasonable. A lot of people get pleasure from giving, as much as from receiving, and it seems quite spiteful to deny them this pleasure.
The OP hasn't said that she insists on a 'thank you', or expects it in a particular manner - I'm not sure where that came from.
When my son was about four he wanted a particular gift for Christmas, and the shops had sold out everywhere, so we told him that Santa was unable to get one, but that we would try to get him one for his birthday in February. He was happy with that, but then I managed to get one in time for Christmas.
My MIL hadn't bought presents for the children at that point, and I let her give my son the present. He was so thrilled, and kept saying that even Santa couldn't get one but Gran had done what Santa couldn't
. The pleasure that she got from that was lovely to see, I was delighted to have 'given' her that feeling, and my son loved his present (something to do with cars bumping into one another). I'm not sure what that says to you about any of us (
), but for us it made for a really happy Christmas all round.