How can the Microcontroller run anything else if the interrupt is really fast?
Short answer is, it can't run anything else if the interrups keep interrupting the normal execution flow. There are MCUs that run main code for one or two opcodes before accepting a new interrupt, and there are MCUs that don't exit the interrupt context at all if next interrupt is pending.

Executing and returning from interrupt takes some time too, so obviously interrupts can't happen faster than this overhead plus the actual execution time of the interrupt routine. Either you need a faster MCU then to allow more interrupts per second, or slowe interrupt rate to process them.

So yes, as you said yourself, this is crazy, and that is why audio processing is not made in ADC interrupt one sample at a time. If you are limited to this, you have the wrong MCU for audio processing. The idea of loop you mentioned is kind of towards the right way because it does not use interrupts, you have more time to fecth and calculate each sample as there is no context switch. MCUs that support DMA go further than this, the CPU can do other things while samples get written to memory without CPU. The DMA controller can be configured to give you an interrupt after a certain amount of samples have been transferred so you can process it in blocks, maybe not in interrupt context but when you have a time slot for it.

How does some commercial speaker do it? Not with a 8-bit AVR that is for sure. It might have a multi-GHz multi-core SoC with a built-in audio processor or a dedicated DSP.