Seems like your problem is basic competency? Move the "can you even code?" question to as early in the filter stage as possible (first 'phone' screening). If you have a lot of applicants, you'll have to do some earlier filtering in the name of time (like on degrees, years experience, "the lucky half"), but don't pretend it's fair or very accurate since both the false positive and false negative rates will be high.
I agree that some coding problem needs to be used to try and answer the question, though with the right interviewer they can answer it without seeing code. The problems you use for that don't have to be at octree-collision-detection whatever challenge, a trivial nested for-loop is fine -- fizzbuzz level is fine. Sometimes you can rely on github or strong internal referral to skip this, but watch out, and anyway it's worth giving your questions to people you're sure will do fine (you've timed at least yourself right?) for the benchmark data and because sometimes they don't do fine, perhaps since maybe your question is too much. e.g. Floyd-Warshall can be done simply with a few nested loops, still I would never give it as a problem and I'd expect nearly everyone I've worked with to flunk it given only the standard hour (which really means 45 minutes).
Some jobs only need basic competence, so you might want to extend an offer if you've been convinced of its presence. At my last job, which ended up being more technically challenging / interesting than my current job, I was hired after posting my resume to Craigslist which led to exchanging some emails and having lunch with the startup founder to talk about my past work and whether I would be useful for his most pressing work. At my current job, I've been part of on-sites where I've established "can you even code?" is "no". Those were costly failures of not having that answered earlier. But we also like to believe we need more than basic competence, so rejections can still occur because of a lack of "testing mindset" or certain "behavioral answers". Only once you fix your "can you even code?" filter is it even worth considering what else you might want to justify an interview pipeline with more stages than a 'phone' screen or lunch conversation.