Cold email / LinkedIn should actually work fairly well for this. If it doesn't, your approach might not be specific enough, the person you're targeting might not be the right person OR your problem hypothesis is flawed.
Re: general approach, I've seen great response rates with a template like this one below.
Hi <NAME>
It looks like you’re an expert on <SPECIFIC FIELD YOU’RE SOLVING A PROBLEM FOR> - I would love to get your opinion on something I’ve been working on.
<ONE SENTENCE TEASER>
I’m not looking to sell anything at this point, but would love to understand if something like this would actually help you with your job. I don’t want to build the wrong thing!
Would you be willing to hop on a brief call with me sometime in the next week? This would only take a few minutes and if we’re onto something, you’d be the first to know about it.
Best, X