Why doesn't searching for the unique name of my website bring the site up in a top position on Google?
No SEO campaign is perfect. So please throw this notion out of the window. Since search engines change strategies and updates come on the average of several a day, it is impossible to have a perfect SEO score at anytime. However, you can have a highly competitive score regardless of the changes if done well.
Without knowing your website name and any terms used, I will have to make some assumptions.
Where a domain name may be unique, there are generally two scenarios: one, the domain name itself is not a recognizable term or set of terms; two, the domain name is a recognizable set of terms. While this may sound obvious, it is worth mentioning because there are two distinct behaviors that follow.
Domain name is not comprised of recognizable term(s): In this case, the only one searching for this term is you. Branding has not been established so search engines may not recognize the search you are trying to make as a branded search and match sites that have used your domain name such as SEO performance sites, domain name to IP address and whois sites, etc. The reason why these sites will bubble up to the top is because of past queries including automated queries and better SEO performance for the domain name.
Domain name is comprised of recognizable term(s): In this case, search engines are not recognizing that you are performing a brand search and separates the domain name into the terms that comprise the domain name. In this case, sites that have performed well for these terms in search queries will bubble to the top. It would be that these sites not only have a successful search history for these terms, but also better SEO geared toward these terms. It would be to your advantage to increase your presence for these terms only if it is reasonable that people should find your site using these terms. For example, (valuetelevision.com) value television are not likely terms people would find your site with, but (affordabletlevision.com) affordable television might be.
When you do a site:example.com search, you can expect your sites home page to be the first listed followed by your most important pages as Google sees them at that point in time. If your home page is not the first listed, do not worry. This is not a problem and there is nothing to fix.
When you do a search for example.com, you will see listed the sites that perform best for your domain name. If your site is not high in the list, this is not a problem and there is nothing to fix. It simply means that other sites perform better for your domain name. Generally people confuse this with poor performance. I assure you this is not an indication of poor performance. It is however, an indication that you do not have enough branding signals that allow search engines to bring your domain name to the top of the list. Branding signals are required along with primary and secondary searches that include keywords along with site name. For example "seo moz" would return moz.com as a recognized brand. This requires a healthy number of the 46 factors (my count) to be present and a moderately high number of brand searches to make happen. Your site would have to be a preferred search destination for a significant number of branded searches as well as have all of it's branding ducks in a row.
When you do a search for example (without site: or .com), then the behaviors that I described at the top apply.
When you state your website uses a unique name, does that mean that the name isn't a word in any language? If that is the case, I'm wondering what type of results occur on the first 5 results pages. If the name is a word in some language, I would expect you would see many pages of results before spotting your website.
If I perform a Google search on the domain name I use for my website it doesn't appear on the first page and it appears after a cemetery with the same name, though it's not a particularly famous cemetery, nor does it seem to have a lot of links on other sites pointing to it, yet it appears higher in the results than my site, if I search only on the name I use for the domain name, though I have about 10,000 incoming links according to Google Analytics, and the cemetery name has a space so its name doesn't exactly match for a search on my domain name.
I don't worry about that and I don't think you should, either, unless your name is a brand name that you expect people to search on. I don't expect people searching for information for which Google or another search engine may return results on my site will be searching on the domain name. If there are results for my site it will be because the keywords they are using to search on will be in the content within pages on my site.