The change is worth it. I made the migration 15-20 years ago and haven't looked back. It's definitely "messier" since there are many tiers you have to learn to get a full stack web application working -- frontend (HTML, CSS, Javascript and frameworks), backend (Java, PHP, Python etc) and finally the database. And build and deployment for all 3 stacks. But I think you might have to migrate sooner or later to this paradigm of programming since the industry has mostly moved already.

It depends on the kinds of applications you want to write. Web has a more limited (but growing) feature set, and its distribution system is phenomenal. Web development can be as clean or as messy as you want, but that's driven more by requirements than anything. Putting text and images on a page can be clean and simple, or you can go overboard and use thousands of libraries to display your 5kb of text.

Web development is the way things are moving - a lot of "desktop" apps now are web apps in disguise (thanks electron). It's worth getting familiar with web development for the job opportunities alone.