One possible solution is that a "wizard's familiar" need not necessarily come from the spell find familiar. Essentially a wizard makes a 'deal' with some entity to act as a familiar - if you want a smarter one, without necessarily having the benefits listed in the spell, consider looking in-character for interesting creatures or working with your DM to come up with alternatives. Additionally, other spells can give access to more powerful creatures which could be entreated to become your familiar (like the Create Homunculus, Conjure Fey, and Awaken spells).
In the monster manual, it states that certain creatures are sought after as familiars (I believe I am remembering text from the Pseudodragon entry), which would support the assumption that familiar as a term does specifically refer to the creature generated by the Find Familiar spell.
I will go on a tangent - the substantive sum of my answer is above. However, consider that familiars might already intelligent entities limited by their bestial form by the spell Find Familiar. Their nature as Fey, Fiends, or Celestials might suggest, as you say, that they should be more intelligent. Note that one could interpret the spell to mean that they can help (advantage on an ability check) you make intelligence checks to recall lore and on almost anything else. I like to imagine, personally, that Familiars speak in conversation through telepathy and allow my players to roleplay as such - though their animal instincts and limited form restrict them.
There is no place in the RAW that says there is a minimum intelligence for knowing a language, and though it is unlikely that DM would allow a familiar to have a known language - Intelligence itself is just a number that determines how much success a creature will have in investigating, recalling information, or resisting certain magical effects. So a familiar, even when operating alone, could make relatively difficult knowledge checks (up to DC16 or 15).
A character is not the sum of its ability scores - if that weren't true, Dungeons and Dragons would not be a roleplaying game. The player or DM that roleplays a character contributes his or her own experience, intuition, and decision making. Things that we consider intelligence in real life often have to do with how people act and whether their decisions are 'intelligent' or not. I think everyone would have to concede that a high intelligence character can still make consistently stupid decisions and a low intelligence character can make consistently smart decisions; and, while I have seen DM's that believe otherwise and relegate all decision making to premade narration based on rolls, I don't believe that is proper roleplaying.
Thus, going back to Find Familiar produced familiars, one does not need to roleplay them as untintelligent beasts. They can be rich characters that contribute to the game, and people shouldn't feel the need to roleplay them as stupid because the number that says how often they succeed on an intelligence check or saving throw is low.