The site is fully indexed by Google (and Bing) and new pages are automatically submitted for indexing. Verification won't directly affect search results.
You would only need to verify the site if you want to access the Google Search Console (to get information and insights about the indexing status of your site).
You are right in saying that I have done this for the e-voice.org.uk domain. And so I can see that google has picked up the sitemap for your site, and has indexed the 10 content pages on your site.
So, if you aren't seeing your site in the results for search terms you're interested in, then this is down to Google's algorithms rather than any technical reason on Voice.
The 'classic' way to improve ranking is to ensure you have lots of good interesting content that is kept up to date, and to get other sites to link to your site.
For more in-depth insights then yes I would recommend getting your own domain name (llebka.org or whatever your preference is, they are all available!), then associating this with your Voice site and verifying it with Google so you can access the Search Console.
You might find that having your own domain gives your site more authority and thus improves its search ranking - but I honestly don't know for sure. I find that when I google for site names they come up pretty high in the search results whether they're using the e-voice domain or their own!
Once you associate your domain with the Voice site, then Voice will automatically create an SSL certificate for that domain for secure https access to the site.