Many new red LEDs are phosphor converted - even if they look very normal.
I got a batch of red 3mm LEDs that have a suspiciously high forward voltage around 2.8V. I first really wondered if they have some kind of defect, but then I shined with a UV-flashlight on them and they started fluorescing. They clearly are phosphor converted, even if they look completely normal.
Many red high power LEDs are also phosphor converted because direct emitting red LEDs get very inefficient with higher temperatures. If you need a red LED that is still powerful at an ambient temperature of >80°C, like in a video beamer optical engine or potentially in the brake light of a car, then it can be more efficient to use phosphor converted red, although it needs more voltage.
And yes, the white LEDs with "classical" cerium doped YAG phosphor indeed have a catastrophic CRI because they principally just mix yellow with blue light. To get a better CRI you also mix gadolinium dopant into the YAG which just broadens the yellow emission into longer wavelengths. I suppose that doesn't influence the decay time. If you want even better CRI, you really mix different phosphor types with different emission maxima.
two 3mm LEDs in daylight:
the same LEDs in UV-light:
the same LEDs in operation: