How can thermal electrical noise be uncorrelated yet appearing at a certain frequency?
If you convolve your U(t) with \$e^{i\omega t}\$ then you'll typically get a non-zero result. This means there is typically some component at \$\omega\$.

Note that if you convolve a delta function \$\delta (t)\$ with \$e^{i\omega t}\$, you'll also get a non-zero result, for all values of \$\omega\$.

The difference between noise and a delta function is the variance in the measured power. With a delta function, the spectrum is flat and consistent. With noise, the spectrum is only flat and consistent after sufficient averaging of multiple convolution results.