Honestly, I think you've hit on something I've been feeling for a while. So many people try to pit these two channels against each other, but they're doing completely different jobs for your audience's brain.
LinkedIn is about showing up and being useful long before anyone even knows they need you. It's that slow, persistent drip of value that means when the pain finally surfaces, your name is the one that floats up first. It's like planting seeds in a garden you won't see bloom for months.
Google, on the other hand, is the place people go when they're already holding a shovel and know exactly what hole they need to dig. They're searching for solutions, not inspiration.
For me, the real frustration is when leadership expects one channel to magically cover both. You can't treat demand creation like demand capture and then wonder why your Google traffic doesn't convert straight away. It's two different conversations with the same person, just at different stages of their journey.