User blog comment:DandyAndy1989/Discussion - Why is Lincoln on every title card?/@comment-32045623-20170519021148

I actually have an answer to this query.

The Loud House is styled after (among other things) the Sunday comics. (This influence shows most in the episodes title cards.) Some Sunday comics, in particular, Garfield, have a unique title box (they were MUCH for creative once upon a time) that incorporate the main character in someway.

I'm almost 100% certain that this is the reason why Lincoln is in every title card, regardless of the role he plays in the episode.