Interpolation and harmonics
Upsampling requires two steps:


Inserting zeros. This does indeed replicate the spetcrum in the frequency domain. For N zeros inserted you get N copies of the original spectrum
Lowpass filtering to remove the mirror spectra. That's often also called an "interpolation filter".


The choice of interpolation filter depends a lot on the requirements of your specific application (signal to noise ratio, spectral suppression, shape of original spectrum, transient behavior, phase distortion, latency, MIPS, etc.). There is no "one size fits all" solution