My rule nowadays is to go with a web application unless you have a good reason not to. The difference between the work done for deployment is enormous, and it's enough to justify avoiding desktop applications 90% of the time.
It starts out as a web app, because you can iterate quickly and its mostly platform agnostic. Then people realise that what you're doing means a 'dedicated' app is a better experience.
At this point, the correct choice would be to have someone create a desktop app that uses your existing web app as an API. If your web stack handles accepts/content-type switching in a decent way, this may mean very little or no work on the backend.
Unfortunately this is also the point where its become common to adopt Electron. Like a lot of things that are wrong in this industry, I blame it on the cool kid javascript/front end developers. "Oh sure I can just make you an electron app for desktop users, its just js/html/css".
Having a desktop application doesn't mean you can't/shouldn't have a web application. It also doesn't mean they need to look/behave exactly the same.
Yes. Desktop applications can be just as quick to develop and iterate on as web based ones. Slapping together a winforms UI in visual studio is still the quickest way to get to MVP status.
A better solution would be something like chocolatey (https://chocolatey.org/), which is basically a package manager for windows. It doesn't do the automatic updating, but doing that with a remote powershell script should be doable.
For some perspective though, if you automate this as part of a CI builds it should be a once off cost, it's not a justification to make something a web app instead.
Managing updates to a cluster of backend servers is at least as difficult, because they usually have shared state (a database).