I had the very same NAD C370 and C542 kit in a prior "B" system (With Totem ARROS and Chord cables) so I know the NAD component strengths and warts very well. IMO your C370 was among the better choices in your current price strata and will best most receivers hands down....full stop.
When it comes to that ethereal system synergy that you are currently lacking, the main non-fixable "wart" against them is their signature NAD sound (you already saw it .....amp AND cd ....) that does not marry well with a not-insignificant number of electronics, speakers, AND cable brands .
The second "wart" against the C370 that IS largely fixable was two mods:
(1) changing the stock uber- crap soldered-in "zip" power cord and
(2) changing the really crummy stock as-cheap-as-you-can-go preamp jumpers.
I had the stock power zip chord professionally dremmelled out in favour of a quality female IEC and external power chord . Big difference.
Also I replaced those ultra-POS cheap metal jumpers with short run RCA terminated interconnects. Again, a step up.
These two mods together helped tame that "dark" overtone and sharpened the overall performance. ....the improvements were not subtle.
IMO..... if you accept a higher price ceiling to say, to a $500-$1000 range for an integrated amp (fuhgeddaboud any receiver with their crummy power supplies and dodgy stats that are already well publicized ) then the available integrated amp suggestion list becomes exponentially meaningful.
The degree of improvement in audio performance in this step-up price point strata that you are craving is not insignificant. These units are all built to a pricepoint and you only get what you pay for.