That might be your character concept.No they don't. You can "participate in magical culture" with a first level magical adept feat. Or the Eldritch Knight can learn as a result of defeat.
My character concept for an Eldritch Fighter is an Elf who grew up in a High culture, where there is a military force that specializes in blending magic and soldiery. This military institution is where the tradition of some elves growing up proficient with sword, bow, and cantrip, comes from. The High elven culture strongly associates the Eldritch Fighter tradition and the Bladesinger Wizard tradition. These are patriotic institutions that the High culture values and makes prestigious.
What needs to be available at level 1 depends on the character concept.
For me it is a nonstarter, to require a feat tax in order to make a Psi Knight a Psi Knight.magical adept feat
Which is why telepathy and telekinesis should be first level feats.
To require a feat tax to make an Eldritch Knight an Eldritch Knight, would make the game unfun for many fans.