Keeping the event live gets you engaged leads who are in consideration phase. They are looking for a solution and that webinar is your golden ticket to Sales. Webinars are a sweet spot for engagement and building value, that not having one would simply not create.
The sign-ups are primarily looking to get content. They are silent, not active and have most probably better things to do than attend a webinar (in most cases, well). The theme of webinar piques their interests.
But to answer the question: What you are essentially looking to solve is increasing qualified leads which is a long process and needs you to get in quality traffic. Maybe a content strategy where you get eyeballs and build up problems. If people are engaged enough, they just might want to be more proactive when you offer solutions. Driving qualified traffic usually involves cutting down on channels that do not align well with your ideal customer. It's standard inbound with a crazy theme to build and solve constantly. Like a cycle.
An extra ebook? Well, if it houses your bigger learnings and is more valuable than the webinar in itself is. Coupons/collateral. Meh. You will just get less but more qualified traffic which might give a bubble effect.
Having said this. It's a hypotheses. Everything is. Whatever works. Just as long as the qualified lead count keeps on increasing.