I guess it's useless to try to trace images this complex (=full of noise) to vectors. At least you should have much more resolution than your linked axample has and you should reduce the amount of the crap radically, Finally: Your engraving system seems to be designed for bitmap images.
Cleaning the bitmap image can be started by making a BW mix where the yellowish texture gets whitened. Photoshop's Image > Adjust > Black and White is for the job. Vector drawing programs have less possibilities for bitmap image fixing.
Several other programs with hue selective adjustments exist. Buy and install nothing by trying web service Photopea. It replicates a good slice of Photoshop's basic functionality.
Curves tool can be used to deepen the blacks and lighten the greys. If your engraving system understands and renders greyshades, this step very likely makes the result worse if you kill too much greys.
At least now the need of high resolution image is obvious.
The random dots and possible missing black pieces must be fixed manually by inserting black and white. Automatic denoise filterings very likely remove also some wanted details.
Multilayer editing program is a delight because nothing is destructive. Even the adjustments can be adjustment layers in Photoshop, tweaking both of them for the best combined effect is possible there. With layer masks you can apply different adjustments to different areas.