In my experience, there is definitely truth to the notion that cold crashing causes the carbonation to take longer. What I eventually realized was, once you take the extra few days needed for cold crashing and gelatin in the fermentor, and the extra few days needed to carbonate as a result, and the annoyance of having to keep intervening on your beer, you might as well just get that beer in the bottle right after it's done fermenting and let it run on autopilot. Let it sit for a week at room temp to carbonate, put it in the fridge, and forget about it until it's ready to drink. In doing so, you're essentially moving the "clarifying" step to the bottle as the packaged beer lagers in the fridge. The overall duration between brew day and drinking a clear brew will be about the same, but you fast track getting the beer out of its attention-needing phase and you free up your equipment (chamber and fermentors) sooner, which enables you to move on to the next brew.
Of course, for this process to work to its fullest you kind of need a dedicated beer fridge that can hold a full batch of bottles for the lagering phase--otherwise you'll have to lager them in batches, which takes a lot longer. It's worth mentioning, though, that the beer will still taste fine without the extended lagering.
There might be a bit more gunk on the bottom of the bottle when you skip cold crashing in the fermentor but there will always be some, and if you pour to a glass it hardly matters anyway.