I have two clients who have Weebly websites. One of them is a well established, highly respected company in our local town. When they first came to me their site was on page 15 of Google for their best favoured search term.
Because of my experience with the other Weebly website I suggested that initially we should promote locally and regionally, and then, if successful, look to get a national listing which is really what they wanted.
By applying basic SEO good practice the site moved quickly to page 1 locally. Soon afterwards with more work on content wording and links we got the same regionally. Now, we are getting a good page 1 listing for some of our key phrases nationally.
Personally, I don't agree that these types of sites cannot achieve the same excellent results as others. It's about content, structure, page element optimisation and utilising some of the excellent tools available to us - particularly from Google.
Quite simply, if you like Weebly, and I know some folks do, stick with it and focus on ensuring that all of the SEO boxes are ticked.
Best of luck