How do you know what's actually working in outbound marketing?

We're doing cold emails, LinkedIn outreach, some cold calling, but honestly I have no idea which channel is actually driving results. Our attribution is messy, people might see our LinkedIn message, ignore it, then respond to an email weeks later. Or they'll book a call after multiple touchpoints and we can't tell which one was the tipping point. This makes it impossible to know where to double down our efforts and budget. How do you guys track multi-channel campaigns without going crazy trying to attribute every interaction?