Just if anyone stumbles upon a similar issue: I accepted Alexis Gatuingt as best answer, even though skdishansachin gave very helpful info concerning broadcasting.

After reading through requirements and counterweighting different arguments this was my evaluation:

• polling: Send a request every 30 or 60 seconds to update the state could be acceptable, but in general terms it produces pretty much network overload. As more of such services are needed it's not very viable and anyway a brute solution. So I better not start with that.
• Inertia Middleware seems a good way inbetween: With every page change some params can be added to the page, and in my actual usecase, page changes or updates are quite possible. Now an OBS: In a SPA (Single Page Apps) and Vue, Inertia middleware is not called until a window reload is triggered, so the param has to be also included additionally on any vue-router change (router.afterEach on global level). inertia middleware by itself would not be enough clean.
• Broadcasting and Laravel Reverb is without a doubt the architectural cleanest option, nevertheless it has a cost in server management, as every client connected via broadcast opens up a websocket, which in bigger apps needs more personnel, knowledge and eventually more server resources. So it may be quite quick costly and and overkill for example for a MVP (Minimum viable product) or for apps with not too many users or if the usecase allows page-changes between (longer calculations)

So in my specific usecase, long-polling is ugly and I want to avoid network overhead specially if userbase grows. Broadcasting and reverb would force to open up a pandoras box with server config, eventually requiring more server resources, etc. and on long term force much faster to have specialist in that field configuring the server when the userbase grows quickly and probably enforce to upgrade server resources (hosting cost).

So with that in mind, and on that specific use case (long running calculation process which is triggered consciusly by user which probably makes page changes while it updates, low payload, few of these UX cases) Inertia middleware and Vue-router guard will my momentary way to go, nevertheless in mind, that it's not 100% clean from architectural perspective, and if more similar UX cases are required, websockets and broadcasting would become more senseful (less calculation time between page changes, clean UX updates which do not require page changes, clean channel for all such UX updates which can be handled more centrally and do not charge the payload between any request). So as long as these status updates, messages alerts do not raise page load times too high, it will be my way to go, until websockets and broadcasting become necessary.