How do you solve the problem that half the country can’t even be bothered to participate once every four years?

I assume you're talking about the US electoral system?? That's very different.

but how would we get people to engage with such a system?

By empowering them.

Consider how the current electoral system disempowers people:

1. Some people literally cannot vote or risk jeopardizing their job taking the day off, others face voter suppression tactics

2. The FPTP system (esp. spoiler effect) and the present political circumstances mean that there are really only two viable options for political parties for most people, so many feel that neither option represents them, let alone their individual positions on policy

3. Politics is widely considered to be corrupt and break electoral promises regularly. There is little faith in either party to represent voters

But, in a system where you are able to represent yourself at will, engagement is actually rewarding and meaningful. It won't magically make everyone care, but direct democracy alongside voter rights reform would likely make more people think it's worth polling.