I think you are looking at the wrong things. You say your targeted page, on your 1-year-old site, is also 1-year-old right? Has that page changed at all during the past 6 months? If it has, then there is your answer.
Search engines do not rank websites, they rank individual web pages, therefore removing an individual web page will never lead to that same page being ranked. If the page you are thinking of removing has content that violates Google’s webmaster guidelines then fixes the page content, there is no need to remove the page, just the offensive content.
In your post, you did not mention what that SEO company did on your behalf. Did that add or change content on your web page? Did they duplicate your content and post it on other websites? Did they add a bunch of competing pages to the SERP page?
Give us a few more details and we may be able to narrow down the possibilities for you.