No, normally it is reached at a much lower Mach than Mmo.
An aircraft at it's performance ceiling isn't at it's upper buffet margin for any reason but coincidence. When an aircraft reaches it's maximum ceiling, given available power and temperature, it's limited in further climb by power. It is not limited by an upper buffet margin. While one mach number may be the magic number for getting to that altitude, that one mach number does NOT represent the maximum speed. A higher mach number will be achievable (assuming one is not in "coffin corner"). This higher mach number's maximum value will be determined by the buffet boundary and will be the upper limiting number.
Simply because the airplane can't climb any more doesn't mean it's reached it's upper limiting value. You're right; the airplane will generally run out of performance prior to reaching Vmo, or such limiting number at which mach effects such as buffeting shall be found. The fact is that most airliners and even many corporate jets simply don't have the power to push themselves to "coffin corner." Certainly at some point they run out of available power and can't climb any more, and certainly a point may be reached where the best angle for climb coincides with or arrives very close to the stall point; the aircrat has reached it's absolute ceiling. This will seldom approximate or equate to the "coffin corner," in which the mach buffet boundary is very close to the low speed buffet (or departure, as the case may be).