They are not supposed to. DnD 5e rules are clear on that point. They can only a learn a very limited number of spells. It would be completely unbalanced otherwise.
Agreed. Cast two spells and attack as a bonus in the same turn is powerful enough.
It would add some spice to that class, but eldritch knights are not pure casters, and wizards are the only ones that could have a spellbook and learn spells without feats, that would make wizards less distinctive.
Eldritch knights already have "weapon bond" and the powerful War magic and Eldritch strike at higher levels, currently not implemented. They do not really need more spells.