-the higher it go, the lower temp it is, and hence the more EGT margin that the engine could archive with out burning the turbine.
That is essentially correct. When operating at the limiting EGT, thrust decreases with increasing ambient temperature and vice versa, at any altitude.
regards,
HN39
P.S. By "EGT margin" I understand you mean the difference between the limiting EGT and that corresponding to the thrust rating. As said earlier, the rating selection is somewhat arbitrary. There is no particular reason for the 'margin' to increase with altitude, on the contrary perhaps.