I too have been happy to sacrifice lumens for a broader spectrum of output. This affects more than just tint. Tint is just the balance between the warm and cool hues emitted by a light source. It does not consider the distribution of those emissions beyond weighing the cool vs. the warm output. So, you could have a light source emitting very narrow bands of light that appears white but is not emitting light in an even pattern like the sun. Why does this matter? If the light source emits yellow and blue, then will a green paint made from mixing yellow and blue to make green look anything but green? No, but there are different ways for an object to appear green. It could appear green because it absorbs everything except for green. A light source emitting only blue and yellow would do a very poor job rendering the correct color of such an object. That is why grass does not look properly green under cool LED light.
Have you ever noticed that your depth perception of such objects seems affected? Our eyes are trying to focus on objects from which they are receiving incomplete information. Everyone agrees this affects color perception, at least in certain circumstances. But I go a step further in saying that an incomplete color spectrum also affects depth perception in certain circumstances. This is one of those difficult to describe feelings that can be experienced the first time you use an LED that emits a spectrum closer to sunlight than any LED before. It's just easier to see what lies ahead walking on a dark path through the woods for example, or in a corn maze (can't wait!) when your eyes are perceiving your surroundings with a spectrum of light more similar to the sun.
So, while filtering can change the tint, IMO that is not really the main shortcoming in most LEDs. The LEDs that I find most pleasing are ones that emit a broader range of light wavelengths. You have to see one to fully appreciate it. <Step down from soapbox.>