But in bitcoin there is no such concept of "pulling" payment from your wallet as far as I know. The closest you get are zero-confirmation transactions which is obviously unacceptable.
What exactly do you mean here? A zero-confirmation transaction still needs to be sent first, and it's not something the receiving party can "pull".
I also definitely wouldn't want to manually make the payment every month. And no Bitpay or other bitcoin-on-a-card services.
I can think of a way without using LN: you could sign
locktime transactions up front. Say you want to subscribe for 12 months, and you want to pay a monthly amount. You could sign 12 different transactions that become valid at the start of each month, and give the receiving party the raw signed transactions.
Each month, they can broadcast one transaction and get paid. This requires 12 inputs in your wallet that won't be used for anything else, so it's not really practical. In case you want to cancel your subscription, you can just move all funds to a different address and the locktime-transactions become worthless.
If there would be a large demand for such a service, it could be included and automated in a wallet, but I doubt there's much demand. I have some recurring payments (loyce.club domain name for instance), but I just pay them manually.
Another argument against automated Bitcoin payments would be the large fluctuations in value.