I wouldn't worry about spiders, they are not racing around aiming to bite people (funnel webs excluded...they are a bit over sexed). The only one you have to worry about for your birds in the US is the 4-5 species of black widow we have (they are black and make cobwebs). Recluses are not dangerous to birds, only humans that react to the bites (and those are over diagnosed).
I happily live with all the spiders around my house, or I take them outside if they are inhabiting the wrong place (the shower...). If you kill them, don't complain about the other bugs around your house!
