Do you need a CS degree? No. Will it help? Sure. It also may make other jobs available to you that require it. But there are a lot of web dev jobs that won’t care.
So, imho, a CS degree will not make you a web dev, unless it is a program focussed on that. But it will expose you to a lot of things that will help tremendously, it will look good to a lot of employers, and it will open up other career paths that you may not have considered yet.
Should you do it? That’s up to you. If I could go back in time, I probably would, but I really dig deep theoretical understandings of things. And I’ve done OK without one.