My first question would be; how well are you doing with your present website? If you have a presence on the web that has a good flow of traffic then it would make some sense to do the best to improve it with the addition of features that might help sell your books. One could assume, since you are hiring a web designer that you might be doing well even with your self described sad website. But if you are not doing that well then you might want to rethink the web designer unless they have some great talent and are offering a great price to construct the website.
The tools on either Weebly or Wix are pretty intuitive and the roughest part would be deciding how you want to present the site. Truthfully though you need to build a presence, which means that you have to be giving the readers something more and that often involves weekly activity.
I have to admit that I don't have much presence on the web; though I do have a blogger account and a Wix account and a Weebly account and at one time I had a domain name with another service. I was doing reviews and sometimes personal rants on a steady basis and publishing on the four platforms with the hope of attracting some audience. Along with that I have advertised my own books and even set up links and since I have published the books also on Smashwords I have used links on Smashwords for my giveaways.
It's true that Weebly and Wix might fold someday, but a lot depends on what you are presently using and on whether you are using WordPress with an outside service or whether you have your own server and a static account at home. If you have the home server and use WordPress then it might be best to stick with that. Otherwise if you are paying another service then your options might open.
If you own your own domain name, that's pretty portable throughout. Otherwise in some cases you can let the outside service purchase the domain name for you, which is what I did, but that means that when you leave you lose control over the domain name until it lapses and is freed up. (My domain name has freed up, but I've not been compelled to purchase it yet, since my web presence hasn't moved much.) If you purchase the domain name then you can carry that wherever you go.
I don't care much for WordPress, but that's based on the sites I've visited that use it. I've never used it on any of the websites I have, but it is an available option. I think that Weebly and Wix tend to push their own interfaces, but I've not done the paid accounts and in most paid accounts that I have worked with the WordPress has been an option in tools. I think that if your are comfortable with WordPress then you should look for that as an option. (Unless you intend to rely on your web-designer for all your changes.)
Using your own domain name is probably more important than which service you use; since once you have a presence, that domain is where people will look for you to be at.
If even with your present website you don't have a presence then it opens things up pretty wide; because what you're doing now might be building the presence you need.
Bottom line for me presently is that I'm using mine as an author's page if readers have an interest and I'm presently trying to build a reader base.
