I not sure about the link, often when I try to follow links here, my old tablet freezes on me. So I might be off base here, but my first thought was maybe her objection is price. If it's a price issue, maybe find a pair of cheaper shoes. Generally shoes are more expensive than slippers. But I don't know the price, so sorry if I'm way off base here.
My next thought was that there might be a communication problem here. If you specify that they will be indoor only shoes instead of slippers, would that help? Explain maybe that shoes are likely to hold up longer than slippers so literal house shoes seemed a good idea? I wear out my socks really fast as well, and I gave up trying to buy slippers as well because I put holes in them after a year as well. I just replace my socks every six months or so now. But it seems to me if you are fine with wearing actual shoes in the house, that seems a good way to help with your sock problem.
Personally, I'd go for the slipper/shoe route first. Then ask for socks as a gift for whatever next holiday or birthday if you do those things. Then you can build a sock collection after having something on hand to save the new socks. Preventative measures, yeah?