I have the same issue on my site I have been using free webhosting at hosting provider. I have mailed to ask this CURL issue with their management and I have got replied that they do not support CURL for cloud backup or other API services. Also they said that if I buy they service and get pro from free account then I will be able to use curl php.
So the issue is simply with the webhosting company you are using and not wordpress. I you buy any webhosting then you will enjoy these services.
I don’t think any free web hosting server which is provide CURL extension for free.
I hope this will clear.
