I'm affraid that the TDA1571 is out of the question. It's rated at 6W per channel...that is what one would use for desktop speakers and would destroy your headphones and your hearing if powered correctly. It could not be powered by a 9V battery as the quiescent current alone is 40ma typically and 80ma max as compared to the LM386 at about 4-5ma with the -1 version rated at 375mW mono.
If you really want to do this project, you have to look at the parameters of the device noted in the datasheets. The video showed construction of a very non-standard use of the LM386 that is at great odds with the recommended configuration of the device for proper and stable operation.
You might want to invest a few bucks in a breadboard and experiment with various audio circuits and expand your electronic horizons from there. It can be fun and the real benefit is the learning experience, to say nothing of the satisfaction of designing your own first circuit.
Good Luck!