(I am hugely editing the question. My initial question was if lowerbounds on threshold circuits say anything about P/NP and it seems that they dont. Irrespective of P/NP its an independently true fact that there exists Boolean functions which are exponentially hard for threshold circuits just that we have never seen them.)
Let me ask my followup question in $3$ parts which I guess are related,
If one looks at the space of all polynomial sized threshold circuits (at constant depth or not) then do we know of any natural complexity class in which they sit? The closest I know of is that the class of depth 3 threshold circuits with no weight restriction are known to be in $NP/poly$. What happens at higher or non-constant depths?
Is there any class of circuits against which we know that there cannot be exponentially hard functions if $P=NP$? (Threshold circuits are clearly not of this type.)
Are there reasons to believe that for each depth $d$ there exists a Boolean function which is easy for depth $d+1$ but exponentially hard for depth $d$? And if such functions are found then would it have any implications (separations) for the other usual complexity classes?