Connotation of "Fool" is perhaps that it is by choice?

You ask for “a word that describes a person who behaves in an unintelligent fashion because of pride or because they believe they are more competent than they actually are.” My only offering is the hyphenated “self-deluded”, which is almost always applied to those whose whose opinion of themselves exceeds their abilities rather than to those whose opinion is less than their abilities.

You also ask if “fool” is appropriate to such a person. In some circumstances, “self-deluded fool” may be appropriate for a self-deluded person, particularly one who does foolish (=unwise and imprudent, rather than mere jesting or comic) things as a result of their belief.

No, you are not right in saying that a fool is a person who deludes themselves about their own abilities and so seems dumb, or rather you are only partially right as this describes just a particular type of fool. A more general definition is needed. OALD provides that definition: "A person who behaves or speaks in a way that lacks intelligence or good manners SYNONYM idiot.

I think there is no single word term fitting your description; you have to rely upon phrases, for instances phrases using "out of".

• a fool out of pride, a fool out of short-sightedness