I do it manually and fully on a desktop Macintosh.
It begins with scanning at high DPI on a flatbed scanner. A mid-range Epson and 600 DPI is more than adequate for old photos, I find.
The raw scan then gets pulled into the photo editing software. Levels are the most important thing to adjust (again, manually). Playing with exposure, shadow level, highlight levels — always trying to get a good dynamic range from the photo (not pulled down so dark you lose shadow detail, not pulling too bright that you clip/bloom the highlights).
I like to try then to remove dust specks, scratches and other defects to try to make my digital version superior to the original. For this step I myself rely on an older version of Apple's Photos app (from like High Sierra?) because the retouch tool is quite fast in that version of Photos (the retouch tool has since fallen way off in speed, usability. I believe however I am missing the Brightness adjustment in this older version of Photos, that is unfortunate).
Adobe Lightroom though is still I believe the gold-standard for photo-editing apps. No doubt they have an excellent retouch tool (but I refuse to pay an annual subscription for software). But there are plenty of other apps that will give you all the photo editing capabilities you might want. I have Pixelmator Pro and Affinity Photo but I have not tried to learn either with regard to restoring old photos. They are probably more than capable. Additionally I have briefly played with apps that focus on B&W (grayscale) images and offer a rich palette of tools that focus on adjusting "tone" and other things I can't quite grok but would like to explore more.
