UK web dev freelancer here. I think it very much depends on the kind of organisations you want to work for. I went solo in 2016 as a self-employed person and never had any trouble accessing clients, who in my case are typically a mix of small digital agencies and small non-technical businesses with no in-house technical resource. I incorporated a company last year in order to firewall my work from my personal affairs, and you may find some companies that will only work with limited companies, but I haven't had that problem so far. If you want to work for big corporations I expect you will run into such issues or the problem you describe yourself. However it's definitely worth approaching different shapes/sizes of business. As another commenter mentioned, you should definitely be aware of the ramifications of the IR35 regulation as this will probably shape the sort of work available to you. Some people work with "umbrella companies" in order to be able to fit better within the constraints of IR35.

All told I've never looked back after ~7 years of doing this, and while job security can obviously be an issue, the comparative flexibility and freedom over your work/life balance etc more than makes up for it. Good luck!