Whether pagination is saved (like ?p=6 in url) doesn't have anything to do with SPA vs SSR. Just like an SSR /search endpoint that uses POST over GET so you can't just save/share the query.
You're blaming SPA but as usual it's just people not caring about polishing application UX.
But in the YouTube case I don't even have pages! But anyways, When not going the SPA route I get all this for free. Navigating the search result pages automatically ends up in the history. 0 lines of Js required. Also since the whole page is generated server side even the scroll position within the result page is restored properly. Again 0 lines of Js. This already worked in the 90s. Today it doesn't, and we call it progress.
We had plenty of sites in the 90s breaking due to using POST on almost every navigation, or embedding a sessionid in the URL that would break the site when being bookmarked/shared
You could build even worse horrors using <form target="..."> to update frames/iframes... partial page updates without javascript. This still lives on in some oracle web applications...
You're blaming SPA but as usual it's just people not caring about polishing application UX.