Can a single, large multilayer ceramic capacitor replace the classic electrolytic + ceramic decoupling capacitor arrangement?
The approach is still relevant today. Why it is so is explained quite well by Dave Jones in this video (EEVBlog #859).

The reason why paralleling bypass capacitors is still relevant is because the frequency response (i.e. actual impedance) of the single capacitors is different due to different parasitics. To obtain a setup that will effectively bypass a large range of frequencies you need to combine capacitors with different nominal capacitance (and/or different technology and/or construction).

This all has little to do with the specific technology of electrolytic caps. In modern digital electronics the power pins of the digital ICs draw sudden pulses of current due to internal circuitry switching. These pulses have large bandwidth and need a bypassing scheme that attenuates the voltage drops these pulses induce in the power rails. 

In other words, if you want to keep the voltage at the rails constant, you need to attenuate a very large range of frequencies, and this is not attainable (in general) with a single bypass capacitor, especially in digital circuitry with high clock rates.

To give further insight: think of a single capacitor as a tank circuit, where the capacitor may resonate with its parasitic inductance. Assuming a simple model where a capacitor has a single resonant frequency, paralleling different caps, with different resonant frequencies, allows us to "combine" the different frequencies responses so as to get a flatter overall response, hence a larger bandwitdh.