Hi there
Redirecting a site depends on a few things:
- Is this competing brand now gone and over with?
- Is the old site relevant to the site you wish to redirect it to?
- Does it compete for the same queries you have?
- Does the new site have a place for old content to redirect to?
If yes to all of the above, redirect it, but make sure that you follow the steps in Moz's Website Migration Guide. It will help make sure that you move the website properly and take advantage of optimization opportunities, such as URL mapping. I would also look into the Change of Address tool in Google Search Console. Also, read through Google's resources on site moves.
I would perform both a content audit and backlink audit as well on the old site - what content can be updated, removed, or consolidated from the old site? What backlinks currently pointed to the competitor's website should be updated to yours, and which links could be removed & disavowed because they are spammy or irrelevant? Remember, if you inherit a website and redirect it, you don't want to inherit their spammy or toxic links, you just want the good ones!
If you redirect the site and migrate it properly, the old site will fall from the SERPs, but your site should capture the vast majority of that traffic and rankings. Especially if your on-site SEO and content is as good, or better, than the old site's.
Hope this helps! Good luck!