I know LWC and now would like to develop my own web application. Is LWC a good choice? If yes, what stack should I be using?
I don't know of any professional web apps (nor would they be immediately obvious), but people certainly have written LWC-based apps (including myself). The default stack for this is a simple NodeJS-based server, and it works well.
npx create-lwc-app and follow the prompts. From here, you can almost directly run this on Heroku, just add a Procfile and fix up the project.json file to include any extra dependencies you need (
npm-run-all if you want to run the API server and static server at once, which is a default script).
If you don't want to go this route, well, you can use pretty much whatever you want. LWC is all client-side, so as long as you can place the files in correct directories and serve static files, you have everything you need. You could use nginx, NodeJS express server, whatever Microsoft is calling a server these days, Apache, etc. Use whatever you feel comfortable using, LWC is perfectly portable across all platforms.
On the subject of professional web apps, both the LWC component reference, and the LWC playground (which is soon to be retired) make use of LWC. Given its legacy, I don't think the component reference is necessarily a from-the-ground-up LWC app. But the playground itself is. I believe it was one of the early initial proving grounds for LWC.
These probably aren't as satisfying as perhaps if there were a bunch of examples of LWC apps in the wild (not built and run by Salesforce). But it felt worth mentioning these.