The short answer is....not yet.
One of the reasons technology is so exciting is that it is constantly changing. It's an eco-system where new, better products are constantly emerging and replacing older ones.
I know we all get excited about the next shiny new object, but sometimes it's better to stick with the tried and true tested solution and be patient and watch for a while.
I hate to use the term but "the devil you know" is often better than trying to use the shiny new software. Whenever I get excited about a new tech, I start to use it but eventually find that 5% of things that I loved about the old tech and could do better. Sure, there's a few new things you couldn't do on the old tech but there's often that one deal breaker.
It took me 5 years of using Weebly before I decided to build a business using it - and only after they rebranded and added the features I needed - like user authentication.
For me, I love the idea of an integrated solution in Weebly where you don't have to have a whole stack of applications working together, but....
as of my wriitng of this email, Weebly's PROMOTE simply isn't as mature as AWeber so it's worth the cost in time and complexity to weave AWeber and Weebly together. For example, PROMOTE doesn't have pop-up forms so you have to integrate with an app like Privy. Well, if I have to integrate two apps anyway, why not use simply use AWeber which is tried and true and has award winning customer service which is the best in the business?
Also, what about tring to put different peopl;e on different email lists based on the product they purchased? That's the kind of cool feature you get with Infusionsoft and some of the super premium CRMs.
So, one day I envision Weebly being yout one-stop-shop so your whole application can be managed through Weebly - your website - your emails - and the only thing you have to do is integrate with a payment processor like Square, but not yet...
So keep going Weebly on those new features and I'll keep testing it out and let you know when I think it's time.