If I understand case (2) correctly, you are connecting JRemote to MC on the NUC via Wi-Fi, and then playing to "This Device" in JRemote, but outputting that via the headphone socket on the phone to input to your Hifi Amp.
In that case, I believe you are using the built-in DAC on your phone, and outputting analogue audio to input to the Hifi Amp analogue inputs. So you could probably get better sound if you had one of those DACs that work with a phone, using the USB connection from the phone to the external DAC, which you then input to the Hifi Amp analogue inputs. Although people usually only do that when they are using the phone away from home, and playing to a set of headphones.
Some phones, with the right drivers or software, can output digital audio via USB, to be input to a USB connection on a DAC, Receiver, or whatever, so that the Digital To Analogue Conversion is done on the most capable piece of equipment.
My question is, are the NUC and the Hifi Amp too far away to connect them by wires? Also, is the Hifi Amp just an Amp, or is it a Receiver with a built-in DAC? Because if they can be connected, you could either use the DAC in the NUC, or add an external DAC to the NUC, and send analogue audio to the Hifi Amp, or if you actually have a Receiver with a built-in DAC, you could send digital audio from the NUC to the Reciever, and use the DAC in there.
Any of the above might give you better audio, depending on what speakers you have connected to your Hifi Amp, and what the Hifi Amp actually is. But the question is, does the music sound fine the way you are playing it now, and if so, will you really notice an improvement?