I start by searching for higher quality images of the logos. Often there is higher quality source material out there. Sometimes there's already a decent sized logo made up just not in the Launchbox DB.
If upscaling is applicable I'll use a program called ChaiNNer with different AI models to get the desired and cleanest result.
I search for any fonts in the logo and replace the texts with the same or closest font I can find.
I usually do painstakingly cut out some part of every logo. Some are easier than others.
Some logos or parts of them are fairly simple and I will end up completely recreating them for better quality.
I will run the finished logo's through Caesium to lower the file size as much as possible without hitting the quality.
Most logos I cleanup end up being a mixture of all of this.
Sometimes it's simple. But usually not. I try to keep them as faithful to the original as possible.
Its something I enjoy doing and can relax with throughout the day.
@seaview59 replied below this with some more great links.