As a DM designing monsters, you can do what you want.
There’s no rule to stop you from implementing the Mythic mechanics without also having legendary actions. You can do as you like when designing monsters
All four published Mythic monsters have legendary actions.
Of the published examples, all Mythic monsters are also legendary: the Tromokratis, Hythonia, and Arasta from Mythic Odysseys of Theros and the Dullahan from Van Richten’s Guide to Ravenloft.