In my experience, having an aged domain is a big bonus for ranking. If you say that there are only less than 30 links to a brand name, you should be able to change the link profile rather easily. If you had hundreds of links to an unrelated text, that could change the way G perceives your website relevance and make it more difficult to rank for your terms.
However, I've also seen sites rank very well from having getting 301 redirects from popular, completely unrelated websites. I would expect your to see something similar to this benefit.
Other factors that may affect the value of domain: length of blank site, change in registrar, theme of last site
So will it help, yes. Worth the cost? Depends on the price. There is a lot of speculation in the domain registration/dropped domains value of SEO and nothing concrete from G, but based on my experience, buying an existing domain > starting from scratch.