Is a video on home page effective way to communicate the value proposition of a solution?

It depends on your project but from my experience its a mix of both. Something like a Loom video (where there is a shortage of you on the side of the screen) then the UI of the product and subtitles highlighted with keywords that may draw people to watch it/listen to it. Assume that most people first see a video and then decide to turn on audio (assuming that we are talking about a video you want to put on the front page of your website). If you are referring to a social media video then it's a whole other story..

It boils down to your branding. Create a clear message and channel it in ways that it speaks with your audience. Stimulate their imagination, emotions, and spark their creativity. Visuals, they sometimes break or make.

Again based on the type of solution you offer I would also suggest considering Animated Promo Videos. If implemented correctly can be very effective and driving engagement.

Hard to say without knowing your product. I think a video is always good to show people around your product, because potential leads can directly dive into your solution and get a better feeling of it.

Depends what you want to PRIMARILY communicate. New Problem: Do you need to educate users about a new problem + solution? An animated explainer video is useful here, but has no business being on your webpage, visitors here already "get" the concept. If they don't you need to explain the concept much more simply than a video. Old Problem, new workflow: Do you need to showcase the entire product workflow? yes a full UI walkthrough video is useful and valuable on your website. Old Problem, old workflow, new UI: Do you just want to show "aha" moments or cool aspects of your UI - ditch the full video and add put several simple 10s gifs.