I do agree with thehorn's position regarding the theoretical side. The amplifier/speaker match is critical to best performance.
Though years back separates were considered better than integrateds, these days that comparison does not always hold true. Especially considering the more modest high end systems, as are your 2 systems.
Regarding your system 1, I would take your speakers and your 304 to a dealership and compare with different integrateds/amplifiers. Best way to hear for yourself. That said, the character of a system can be altered to a greater extent with a change of speakers than the amplification.
Though years back separates were considered better than integrateds, these days that comparison does not always hold true. Especially considering the more modest high end systems, as are your 2 systems.
Regarding your system 1, I would take your speakers and your 304 to a dealership and compare with different integrateds/amplifiers. Best way to hear for yourself. That said, the character of a system can be altered to a greater extent with a change of speakers than the amplification.