Google provides 2 different search engine services
free & paid
The free one will include Google results and Google Ads
The paid will be dedicated only for your website and you can add your own Adsense account and get paid for that.
The benefits of have Google Search Engine rather than your own one is:
- Adsense account
- Better Indexed Page
- Faster Search and can be called CDN search
- Somehow better SEO service
- increase your conversions
- Customer trust & satisfaction
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I think that it's a bad idea to put a google search box etc on your site to attract visitors. Try using natural SEO techniques like using legit content (not copied) and nice user friendly design etc. Users are always attracted to the content in the site..
I'm not sure I understand what the OP means. If you are referring to making a web page that is completely black with the exception of a Google search bar in the middle of it, then you are wasting your time and money. It will not get popular. People use Google, not other Google-based clones.
If what you meant was to replace the search function on your site with the Google search function, than that would probably also be a bad idea because instead of taking users to the page of your site, searched keywords will take them to a Google page containing a link towards your page, so it would involve extra-clicking.