Hi Julian,
I do not know the details of the circuitry used for motor control in the Source so any comments here are hypothetical and based on experience acquired by upgrading the power supplies of a very wide variety of electronic equipment including many record decks with DC motor control systems.
Firstly the proposed DC power supply shown by John Gordon is very simple and has limited supply line rejection and load regulation. Replacing it with a well-designed regulated supply should provide a noticeable sonic improvement. The reason for this, irrespective of whether there is an on-board regulator supplying the control circuitry, is the additional barrier to supply line interference provided by this regulated power supply. Also feeding the on-board regulator with a regulated voltage will render it less susceptible to input voltage variations caused by dynamic load current swings as well as protecting it from over voltage when the mains voltage rises. So you should aim for a high quality regulated master power supply for these reasons.
As Mark has stated the overall improvement to sound quality will be limited by the performance of any on-board regulation on the motor control board, so for the best result you will need to replace any low quality an-board regulation with something of better quality. This applies to any item of electronic equipment. Improve the power supply, improve the performance.
Using an LM317 may provide a sonic improvement in some areas of performance and will cost little to try. However the overall performance will not be great. I use an LM317 based charger to charge my car battery but would never use it for any audio related power supply. In fact I stopped using it for audio power in 1980 when, encouraged by others, I found better solutions for audio power supplies and have progressively developed them since. The LM317 is an old design and was designed in 1970 for general-purpose industrial use. It has no place in high end audio.
Regards
Paul