First my take. Intel needed a way to be competitive with AMD in multicore benchmarks and performance scenarios but were limited by space and power due to foundry limitations, so they created E-cores.
AMD wanted more cores with less power, so they reduced the speed to allow for more cores in a small space, and a few other changes, but essentially kept all the same capabilities, like avx-512.
My opinion is that AMDs solution is much easier to implement and much more sane and capable in todays world. Please give your thoughts, and this discussion is more about methods than companies.
You got half of it correct. AMD and Intel are using c and E cores for area efficiency. But they are not using them for power efficiency. P cores and normal size Zen 4 cores are more power efficient at iso-performance than their "full size" counterparts. Well, to be totally honest this is definitely the case for Intel, for AMD efficiency is probably about the same since the architecture is the same, but the c cores are smaller so the only advantage of using them for AMD is area efficiency.
Due to the nonlinear nature of the v/f curve a big (in terms of die area), wide (in terms of architecture) CPU running at a low frequency will generally be more power efficient than a smaller physical core running at higher frequency to achieve the same performance. But of course die area is expensive so we have c and E cores.
AMD added the c cores for the exact same reason Intel did. Most software doesn't utilize more than 8 or so threads so there are decreasing gains to adding more full size cores. Instead of designing an entirely new architecture AMD reduced the physical size of Zen 4, which required a decrease in clocks. This way they got quite a bit of additional multicore performance for less die area than would have been required with "full speed" Zen 4 (larger) cores.
It's all a delicately engineered balance of price/performance/efficiency that both companies create. By creating the c cores AMD has validated Intel's hybrid approach.