The gift card concept is a good idea, but I think it would make sense to sell subscriptions (via digital code) on Amazon.
Mullvad VPN does this, and it works out well from a privacy perspective. Since Amazon is the merchant, Mullvad has no idea who the purchaser of the code is.
The purchaser just buys it off Amazon, plugs in a code, and uses the VPN service.
Selling one-month subscriptions on Amazon would make it easy for someone to use Kagi somewhat privately, or to just try out the service without committing their credit card to it. It also makes it easy to gift Kagi subscriptions to friends and family.