Marpa,
There is more than 1 question here inherent in what you are asking as far as I can see. As an aside, I've also owned the P-03U/D-03/G-0s combination and later P-03U/D-03/G-03x, both with very good results.
The 1st question is whether a later clock than the G-0s would serve you well; the answer to this is very much yes, regardless of manufacturer. The G-0s was Esoteric's first commercial product introduction around a Rubidium oscillator-based word clock. It features 75-outputs and frequency word-clocking at a more precise rate that onboard PLL-based designs of the time able to pump out 44.1 and 48.0 frequency baseline and multiples for audio as well as video. It could also put out a 100Mhz 'master' clock pulse as well. It has an additional feature where you could drive it with a 10Mhz 'master clock signal to provide atomic clock sync if you happened to have a more expensive cesium-oscillator atomic clock just laying around (in those years)
. It was a very good unit and all its connects are 75-ohm as you many already know. The G-0s had higher phase noise than today's Rb and OCX word and master clock units and therefore more jitter. In short, yes, by moving away from that clock you could achieve a more musical result assuming you picked a modern clock (Esoteric, DCS or Cybershaft or others) with typically lower phase noise and better Allen Variance stats that the G-0s had way back when. It's oscillator has also aged quite a few years and unless it's been back to Esoteric, it's precision is also not quite what it should be due to the time that has passed.
The K1 you have is spec'ed ideally for 50-ohm input while the G-0s outputs 75-ohm. According to the Esoteric's specs, the K1 can take in 75-ohm but I think it is more ideally suited for a 50-ohm pure connect so getting a clock that outputs 50-ohms (your 2nd inherent question) is a move you'd like to make and then making sure you have a proper 50-ohm cable. This does make a difference according to the testing mentioned here by several audiophiles and 2 manufacturers (Shunyata and Cybershaft).
Which clock....the 3rd question underlying your post is 'which clock'; I happen to love the OCX-based ultra low phase noise effects of Cybershaft 10Mhz master clocking with a proper 50-ohm cable. The Esoteric G1 and G-01 are also in this league but Rubidium based, the G1 having pure 50-ohm outputs so that (G1) is another viable option for you to explore.
Overall my response is, 'yes' make the switch for the above reasons. My preferred configurations for you to think about are;
1. Cybershaft OP13 or better Ultra Premium (may have to wait for it) 10Mhz Master Clock with Shunyata Alpha or Sigma "CLOCK50" cable
2. Esoteric Grandioso G1 10Mhz master clock, with Shunyata Alpha or Sigma level "CLOCK50" cable
Don't mix 75-ohm cables where 50 are called for no matter what your read; it does make a difference!
Need to keep this somewhat brief as I'm working this response in while on the job....feel free to IM me with a phone# and I'd be happy to give you a call and talk live.
Hope this helps....