Answers
Why do some apps not work on school Wi-Fi?
Because the network blocks them by domain or by category, usually under a filtering policy the school is legally required to have. It is a network decision, not a fault in the app.
US schools receiving certain federal funding must filter internet access, and most implement it with a commercial category-based filter. Social media and messaging are common categories to block.
The filter operates on domains, so it catches a website and an app equally. A browser-based tool is not a way around it and should not be treated as one.
What a browser-based tool does solve is the separate restriction on installing software, which is why it works on the device even where an app cannot.