I can’t see the Kef speakers being the issue. I suspect the amps and processing are not up to much, with insufficient power, and more specifically insufficient power available for the low frequencies.
Power figures on these multi channel amps are often stated in quite misleading ways. Each channel of a multi channel amp might be optimistically rated at say 100W, but there’s no way the amp would drive all the channels to that power level simultaneously. This is an active system (amps driving the drivers directly with a filtered input signal to the amp, not passive filters/cross-over separating a full range amplified signal). Active systems are in many ways better, but the amp power needs to match the driver requirements, else it might end up being worse than a passive system with less channels, but each with higher power.
Low frequencies need A LOT more power to keep up with the higher frequencies, and that is only gonna be made worse if the low frequency cabinets/drivers are not so sensitive. It seems Kef low frequency section is more about accuracy and quality, not quantity with cabinet loading to make them loud (maybe not so highly efficient).
The overall sound level is likely going to be limited by insufficient power to the low frequency section, whilst the high frequency section might have more amplification available than it needs (20W on a tweeter and 50W on a mid, might need 150 or 200W on the bass, to provide a balanced sound at high level for example). If you had a multi channel (10 channel in the case of the Emira, as I understand it), then it would be normal to use such low power amplification for the mid-bass and higher frequencies only, but have a dedicated higher power bass/sub-woofer amplification, and that would work well.
560W could easily be a real world 150W total, but that 150W might be split into sections where more power than is needed is available to the tweeters for example (when they don’t need that much) but only says 60W available to the low frequencies, when that just isn’t sufficient to get to a high enough sound pressure level to match a typical high end car system.