Why doesn't mass show up in the stress-energy tensor explicitly?
What makes you think it doesn't?
The stress energy tensor describes the flow of energy and momentum through space-time. Multiplying the stress-energy tensor by the velocity (4-velocity) of an observer gives the energy and momentum contained within a unit volume according to that observer.
"Relativistic mass" is another name for energy, and is one component of the stress-energy tensor, so in that sense "mass" could be considered to be one part of the stress-energy tensor. But it's not the whole tensor - since momentum and energy are intertwined in a similar manner to space and time. Energy is thus not a tensor, it's one component of a tensor. Mass in special relativity, in the sense of invariant mass, is defined as the Lorentz invariant length of the energy-momentum 4-vector - of an isolates system or particle. The "isolation" aspect is sometimes not stressed, but if you read the fine print in say, Taylor & Wheeler's "Space-time Physics", you'll see that it is assumed that one has an isolated system or an isolated particle when one talks about the invariant mass of the system or particle. If the system is isolated, and in flat space-time, one can find the invariant mass of the system from the stress-energy tensor by integrating the stress-energy tensor to find the total energy, the total momentum, and using the relationship E^2 - p^2 = m^2 (throw in factors of c as needed, if one is not using units where c=1).
If one does not have flat space-time, one needs a different concept of mass. The details start to get technical here, I'll just mention that one might use the ADM mass, the Bondi mass, or the Komar mass, if one of them happens to apply. For some situations none of them apply.
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