Yes. The random generator has a Category filter for exactly this. You can limit the pool to Legendary only, Mythical only, or exclude both entirely.
Step 1
Go to the Random Pokemon page.
Step 2
Find the Category dropdown. Options are: All Categories (default), Legendary only, Mythical only, and Non-legendary.
Step 3
Pick "Legendary only" and the pool shrinks to just Legendary Pokemon. The Generate button counter updates to show how many are available. You can still add region, generation, or type filters on top of that.
Step 4
Pick "Mythical only" for Pokemon like Mew, Celebi, Jirachi, and the rest. Mythical and Legendary are separate categories.
Step 5
Pick "Non legendary" to exclude all Legendaries and Mythicals. Good for challenge runs or randomized playthroughs where you want only regular Pokemon.