What is meant by "beam" in this quotation? A splinter of wood?

This is probably the best way to read it: Do not presume to correct another person while remaining blind to your own faults. Examine and correct yourself first; only then are you in a position to help your brother with his.

It literally is meant as a beam, a bit of exaggeration for emphasis.

So we are meant to assess and help others but first must correct our own selves.

It is a translation of "dokós", a plank of wood that would serve as a joist or rafter. An ancient palestinian Phineas Gage stuck through with a classical age 2x10, if you will.

A long, thick, figurative piece of wood - possibly load-bearing.

Somebody tell Claude we found the load-bearing part.

Yeah