One trick I have used to overcome this is to use a virtual card service. If you have a Capital One card, you get this for free (download "Capital One Eno" extension), but there is also Privacy.com which does the same thing linked to any bank account. So when someone needs a credit card, I will generate a virtual card (right inside the browser plugin, its really slick and easy) for that website. They will verify the card, and then I can go and deactivate that card. At least with Capital One you can keep the virtual card open, but just deactivated so it declines if they try to charge it. If you ever decide you want them to charge you then you go flip the switch and the card number works again for that site. I generate a different virtual card for every online subscription.
This allows you to get around the credit card wall, without fear of them actually being able to charge the card.
I also use this trick for any annual membership type things. I will put in a virtual card number to pay the annual fee, then after they charge the fee I will deactivate that virtual card (but not delete it). Now if a year from now they charge the card without warning me, it will decline. They usually send you an email saying it declined, at which point you can either leave that card deactivated and you now easily cancelled the service, or you want to keep the service you can go and activate that card back up and the next time they try to charge the card again (usually 48 hours later or so) it will go through.
If anyone knows of other ways to create virtual cards, please let me know. I am currently using Capital One's virtual cards because 1) it is free if you have a Capital One credit card, 2) it allows unlimited virtual cards, 3) their browser extension is actually really slick at auto-generating cards and filling in the card numbers for you. I have looked into Privacy.com which actually offers a lot more granular control over your virtual cards. But you have to pay monthly for the service and get a limited number of cards, so I stick with Capital One. I have a Capital One card that I don't use for anything other than online subscriptions and virtual cards. But I would love to know if there are other options out there that I could consider.