Hello Andrew,
Few things I need to know before answering your query.
1> Is your old site penalized earlier or have low quality backlinks if yes don't do redirects it won't help in fact that will harm your website.
2> if your old website is completely fine I mean no penalty/ no low quality links then it is good practice to do 301 redirect.
3> as you mentioned in your query that your old sites niche is different then current that means if you will redirect that will not serve any value to user and bounce rate will be high and I doubt boost in ranking.
Hope above suggestion helps you in deciding what would be the appropriate steps.
thanks