Can I use the VS Code API to implement language features in a web extension?

Question

I generated a VS Code web extension, registered the language into the package.json file, and put this code snippet into the extension.ts file:

console.log('activation');
context.subscriptions.push(vscode.languages.registerHoverProvider(selector, {
    provideHover: function (document: vscode.TextDocument, position: vscode.Position, token: vscode.CancellationToken): vscode.ProviderResult<vscode.Hover> {
        console.log('hover');
        return new vscode.Hover('test');
    }
}));

If I start the extension by pressing the F5 key (it starts a new VS Code instance to run the extension), it works fine, but if I run the run-in-browser npm script it doesn't work. The activate function runs in both cases, because I see the text 'activation' in the console, but if I run in the browser I can't see the text 'hover'.

In normal (not web) VS Code extensions I can use both the VS Code API and the Language Server Protocol to implement language features like hover, formatting, go to definition etc. In the web extensions page they don't mention that I can only use the Language Server Protocol (and not the API) in web extensions, but in the only web extension sample where they implement language features, they use the Language Server Protocol.

So, is it possible to use the VS Code API to implement language features in web extensions, or I have to use the Language Server Protocol? If it's possible, why my code doesn't work in the browser but it does in desktop VS Code (using the web environment)?