By theory wouldn't this bring my site to the same position if I am optimized for the same keyword phases?
If your site was precisely the same age, with the same kinds of content and about the same organization and number of pages, then yes, it would be a horse race. But of course that is never the case.
The thinking behind the tactic you mention is simply that if those sites are linking to your competitors, and you have an equally good or better site, then those sites may also be interested in linking to you. It's more an SEO 101 starting point; a basic base to cover.
The trick then is to create more/better content, and market your site more effectively, and listen to your user base. Also ask yourself two questions: 1) What are your competitors doing that you could do better? 2) What are your competitors not doing, that should be done?
Cleaner you should make sure that the links you get are only the ones that are relevant to your niche
You should also get some unique ones but this will only bring you to the a same back links you would also the text on the anchors and around the anchors to be optimized. On a highly competitive search term allinanchor wouldn't be enough but for most average med serps maybe..
Do search for allinanchor:keyword
these off page are only going to give your the rank if you have equal or better on page factors