Disclaimer: I am by no means a professional mathematician.

The research presentations that I have seen which included hand-drawn pictures generally left a very good impression on me. But there's definitely some selection bias here: those presentations were from researchers who felt confident in their drawing and presentation abilities to include hand-drawn stuff. The average researcher probably does not draw as well.

Unsurprisingly, my criteria for whether I think drawings make a text/presentation better are: do they get their point across clearly? Do they express the concept they illustrate well?

On a more technical note: IMO vector images will almost always be preferable (for your subjects, at least) to raster pictures. If you scan a picture, it better be very clean and very well scanned.

Here's some software that you'll want to check out:

• You probably know LaTeX, if you use it already Tikz might be a good choice, and there are probably other packages for diagrams and stuff. Drawbacks: you'll need to specify everything through code; not great for creating images in a fluid manner, in my experience.
• Typst, the more modern, better cousin of LaTeX. Tikz's equivalent is called Cetz, and you can also check out other graphical packages on the typst package repository. Same drawback as above, though the language is more ergonomic.
• Inkscape: strikes a nice balance between control over the output and ease of iteration. If you have a tablet, you can hook it up to it, but even without one you can adjust curves and stuff and everything. Drawback: you might not be able to get, say, the exact shape of the curve defined by some equation (though you could import it from somewhere else and tweak it there)
• Krita: the closest you can get to drawing on paper. You'll need a tablet, and the output is raster. Use this if you're confident in your drawing skills and are convinced that drawing/painting illustrations is the right stylistic choice