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"Progress doesn't come from early risers – progress is made by lazy men looking for easier ways to do things." Lazarus Long
I've been working in the desktop development space for years and I wonder, what exactly do I miss? Anyway, are you talking about technologies or the space? Do you strive to write your desktop stuff with web languages, or do you explicitly want to do web stuff?
Fossils last for eons. New languages, frameworks are like Mayflies. Here today, gone tomorrow.
They don't pay you because it is fun. They are paying you so you will come back and do it again tomorrow.
Newer isn't better, just different.
If you are that bored, develop a product in a new language on your own time and go into business for yourself, leaving your current job. You might not be successful with it, but you certainly will not be bored.
I have been better, but I have been much worse, as well.
Yes, you must get experience with those new technologies.. if only for future job security. Since you don't get to do them at work, do your own projects at home. If you make them open source, or at least make the source visible to everyone, you can use them as an example of your work to a prospective employer. Once you have enough years experience with successful web apps that you're considered a senior developer at that stuff, you should be able to switch jobs.
Do you ever get brand new projects? If so, look into using Electron, which is a desktop HTML/CSS/JS stack. Or better yet, use Python. Python is easy to learn and is a great general purpose cross-platform technology that be used to build just about any type of software.
Besides studying in the evenings, think of a project you can do in your spare time. A website for some group you belong to, or build some NuGet packages from some functionality you can think of, then publish them. Build some utility app that you can sell or distribute, or use as the basis for an article on CodeProject.
This will give you some accomplishments that you can then show your boss to prove you are ready for new tasks.
What is good eventually turns bad; what was bad eventually turns good ... repeat.
The Master said, 'Am I indeed possessed of knowledge? I am not knowing. But if a mean person, who appears quite empty-like, ask anything of me, I set it forth from one end to the other, and exhaust it.'
― Confucian Analects