I think you are imagining a problem that isn't there.
It is true that LEDs use several materials, some of which are rare. But incredibly small amounts of these materials are used.
The most common type of LED today used is the blue gallium nitride LED. White light is produced from it using a phosphor material that converts blue wavelengths to longer wavelengths. Common phosphor is cerium-doped yttrium aluminum garnet (containing yttrium, aluminum, oxygen and very little (almost no) cerium).
For example, a common LED that can be run at 8 watts is Osram P9. Osram doesn't specify the dimensions of the chip, but based on the datasheet I can estimate it's at most 9 square millimeters because larger chip wouldn't fit to the final product.
So we can say LEDs need about 1 square millimeter per watt.
Gallium nitride isn't actually used for wafers because a good production method to make gallium nitride wafers doesn't exist. So actually the wafer (most of the material) is sapphire or aluminum oxide. Both aluminum and oxygen are very common materials. Then a gallium nitride buffer layer is grown on the sapphire wafer, and the active layers are grown on the buffer layer.
According to several searches on scientific papers, the buffer layer thickness of GaN is about 20 micrometers. So one watt has 20 µm x 1 mm x 1 mm of gallium nitride. That's about 0.123 milligrams of gallium nitride per watt. 83% of that is gallium rest being nitrogen, so we can estimate that about 0.1 milligram of gallium is needed for 1 watt of LED.
Gallium doesn't occur in rich deposits but at least 1 million tons of gallium are available in bauxite and zinc ores (according to Wikipedia).
This means that our gallium deposits can be used for 1016 watts or ten petawatts of LEDs.
I have around 700 watts of LED lights in my house, car, bikes, etc. And that's a lot -- I really enjoy lots of bright lights. If every person had the same amount of lights, and if world population stabilizes at 10 billion, we could make about 1400 iterations of these lights for everyone. If one iteration is good for 3.5 years, that means we are good for at least 5000 years.
There are far more limiting resources that become a problem quicker than in 5000 years, such as phosphorus reserves used for fertilizers running out.
I also searched for a typical thickness of YAG used in LEDs. One paper said 75 micrometers as an example thickness. At this thickness, 0.342 milligrams of YAG are needed per watt. 44% of that is yttrium so about 0.15 milligrams of yttrium is needed per watt of white LED. Yttrium oxide reserves are half million tons and 78% of that is yttrium so we have about 0.4 million tons of yttrium.
This is enough for 2.7 petawatts of white LEDs. Assuming every person on this planet would use 700 watts of white LEDs, that's enough for 386 iterations of lights or 1350 years if each iteration is 3.5 years.
The other materials are used in such small quantities that they don't matter. For example the cerium doping in YAG is only doping, so very small amounts of it are used.
Also both yttrium and gallium can and will be recycled, unlike fossil fuels which are used once and then they're gone.