To provide an update on this issue: I applied for a physical Yoomoney MIR card to be sent to my home address in the UK. The card took four weeks to arrive. Of those four weeks, the card spent three weeks sitting at the Russian postal service and then in the fourth week the card finally wended its way to the UK (I think most of that final week it was sitting at a depot for incoming international mail).
I know what was happening to the card in transit because my Yoomoney account informed me when the card was still in Russia and when it had left. During the last day or so I then got updates from the UK's Royal Mail. The card was sent recorded delivery so I signed for it on arrival (I in fact requested Royal Mail to keep the package at a local post office and picked it up there). Quite impressed with how smooth and transparent this process was, though it took four weeks.
Even before the physical card arrived, I have been using the separate Yoomoney virtual card to make purchases in roubles ahead of my trip to Russia. E.g. I paid for my hotel this way. I already have an existing Russian bank account so I funded the Yoomoney virtual card with funds from that bank account. I have not tried funding the card from outside Russia. Apparently, there are ways to do this but it sounds complicated.
I was surprised by how easy it was to complete the KYC verification for Yoomoney. It is certainly possible to do this from abroad, simply by taking a photo of one's passport and a selfie. They will accept an international phone number, so you do not need a Russian one.
My main advice to anybody trying to complete the KYC verification is DO NOT USE THE YOOMONEY APP TO TAKE THE PASSPORT PHOTO. This was impossible when I tried (there was no way to align the passport to the app's satisfaction). I almost gave up, but I eventually discovered that if I instead uploaded a passport photo I had taken myself, the app was perfectly happy to use the uploaded photo for the passport details.
I was unable to get my phone to accept Yoomoney Pay for contactless payments. Possibly my phone is not compatible (it says on the Yoomoney website you need a smartphone "that has been officially released in the Russian Federation"). But I am not bothered by the inability to set up Yoomoney Pay as I prefer using a physical card anyway.