For some reason, am I not able to pause my membership just because I’m paying through Itunes/Apple 
I can not believe I don’t have the same possibilities just because am using another platform!
Does anybody have any experience with this and how to solve it?
You should be able to do this through your Apple account. There’s a manage subscriptions page on your account in Apple and you just select Zwift and cancel it.
As far as I am aware pausing and cancelling are effectively the same. The only difference seems to be that with pausing Zwift will automatically start charging you again after the pause period (which is only 2 to 8 weeks, any longer than that and you have to cancel your sub). With Apple you should be able to restart your subscription when you want to resume Zwifting.
I think the pause option is not available due to the way Apple works. Only the user has control over the Apple payments, 3rd parties like Zwift do not have the authority to stop or start Apple payments.
If you subscribed through iTunes, you need to pause or cancel through iTunes. More information is on our Support Hub.
If you need additional help with your particular subscription, please get in touch using the Contact Us form on suppport.zwift.com. We don’t handle specific account-related matters, especially ones related to your money on this public forum for privacy reasons.