I think a better use of your time would be to interview the users as opposed to testing them, but before you even do that I would define who your users are by mental models. You need to distinguish between where users that you are targeting are in the journey.
If they are fresh and never heard of you before, they may not prefer the other website, they just don't know about you, so maybe it's your marketing and nothing to do with the product you made or the competitor's product. As some stated above, it's not necessarily a UX problem. So you're messaging would need to adapt for this type of user to even sign up. So then you would go test that messaging.
If they are an existing user, and you know that they switched to a competitor then that's the person you really want to talk to as to why that it is. What you need to find out is why they choose to stay at the competitor and either replicate what the competitor does or do it better than them and incentivize these people to come back over.
Also many or some users may be shared between both of your sites, so "more" is a relative term if the same users sign up for both and are just using a "shotgun" approach to finding a job. They may not actually prefer one or the other. So this is like your "indifferent" archetype. Defining your users first into thinking modes like this will help you construct more targeted testing later. You can always improve your product but you should know whom you are improving it for and why before aggressively testing.