Let's say I buy a website name (.com) and compete against other low authority websites with the same name but different domain (.co), and then manage to get to the top. Would it make sense for the low authority website to try and compete to beat me to get to the top?
It almost never makes sense to try to compete with the same name as a competitor.
You want to build your own brand. The only way to do that is to have a unique name. You want users to be able to type your name in and be sure they get to your site. You don't want to have competitors in the search results for your name. You don't want users to forget which URL belongs to you. You want your brand to be as obvious as possible so that your (good) reputation is tied to it and isn't used by a competitor.
Brands are so important that you may even want to trademark your brand name. That gives you legal rights when a competitor tries to use your brand name. That allows you to take them to court and get monetary damages.
If you are using the same name as a competitor, there is the chance that the competitor already has the trademark and they could sue you. You can check if they registered the trademark. Even if they haven't registered it, they can argue they have been using it and they have rights to it under "common law". So in addition to not being able to build your brand, you could get sued by your competitor by using their name.
Pick a unique and brandable name. Your website will be much better for having done so.