How to get a page’s featured image through REST API?

Maybe because the custom REST field (fimg_url) was only being added to posts of the post type and yet your request was for Pages, i.e. posts of the page type.. so try changing the register_rest_field( array('post') to register_rest_field( array('post', 'page').

See https://developer.wordpress.org/reference/functions/register_rest_field/#parameters.

But actually, you can use the _embed parameter to have the full media object be embedded into the response of your post/page request.

Example using JavaScript’s window.fetch:

fetch( '/wp-json/wp/v2/pages?_embed' )
//fetch( '/wp-json/wp/v2/pages?_embed=wp:featuredmedia' ) // This also works.
    .then( res => res.json() )
    .then( posts => {
        posts.map( ( post ) => {
            const media = post._embedded?.['wp:featuredmedia']?.[0];

            // Default sizes, which are also the keys in mediaSizes are: full,
            // medium and thumbnail.
            const mediaSizes = media?.media_details?.sizes || {};

            console.log( mediaSizes.thumbnail?.source_url, mediaSizes );
            // Hint: Just log the `post` and `media` to see what properties are
            // there.
        } );
    } );

The above can reduce the number of HTTP requests, but it can also increase the response size, so you might instead want to use the featured_media property to request the media object via the GET /wp/v2/media/<id> endpoint. E.g.

function getFeaturedMedia( mediaId ) {
    return fetch( '/wp-json/wp/v2/media/' + mediaId ).then( res => res.json() );
}

fetch( '/wp-json/wp/v2/pages' )
    .then( res => res.json() )
    .then( posts => {
        posts.map( async ( post ) => {
            const media = post.featured_media ?
                await getFeaturedMedia( post.featured_media ) : null;

            // Default sizes, which are also the keys in mediaSizes are: full,
            // medium and thumbnail.
            const mediaSizes = media?.media_details?.sizes || {};

            console.log( mediaSizes.thumbnail?.source_url, mediaSizes );
        } );
    } );