I accidentally broke this leaf on my philodendron billietiae about six months ago. It seems to be healthy, but it looks unsightly. Can I repair it? For example, can I use plant glue and a toothpick to stabilize the leaf?

Its not fixable. Cut that part of the leaf off.

Why do I need to cut it off?

Because you think its 'unsightly'

You don't. Its your choice. You feel it's unsightly - its all about aesthetics. If the broken end is still green, it is helping support the plant but if you don't like it, there's no reason to keep it.