Absolutely, every component in a circuit adds noise, my understanding is the MDRT adds less noise than the original tanny MJMP will know better why (my guess (partly from pics of them) is coz it's 2 trannys in one but separated ?) so rather than the various taps off the coil all in one. Not that I'm an expert in transformer design but wrapping more coils of copper around where the inner taps come out must have some effect.
Also the tranny "is" the signal in that it provides the Eddy electrons into the circuit(s) that go on to make/amplify the guitar signal. This then becomes a gain structure conversation. When you add gain, it boosts the signal and the noise. The better the signal to noise ratio of each component, the less noise is boosted along with the signal for the same output. So the idea is to have as much signal as the component(s) in the circuit will take without clipping/distorting and stay as far away from the "noise floor" as you can. A lot of noise can be reduced by just getting your gain structure right/optimal.
However, when you exacerbate the situation by using tubes to create distortion (not the original intention of a tube BTW) which is basically overloading/clipping/compressing each gain stage (2 in each 12AX7 so 4 in all, or 6 in 3TM), the noise is boosted even more compared to the signal (it (the noise) is not clipping/compressing, just getting boosted/gain). So the lower the noise floor of each component, the better. Various components that do much the same thing in the circuit can have very different signal to noise ratios/noise floors (for a various reasons). Generally better S/N components cost more. There's an interesting post on here somewhere where he swapped out all the opp amps in his MP1 selecting only the best he could source (freq response/noise floor). Reported result was amaizing reduction in noise and total opening up of the sound.