Have you gotten a warlord swarm book to work? I had a decent one going but it dies to Aviary. Grunts can't handle birds and Slingers are even slightly worse then birds. Basically +1 mana cost for +1 hp which isn't worth it.
It depends what you consider swarm. If you consider swarm a mix of level 1's and 2's (grunts, slingers, butchers, alchemists) then yes I believe I've gotten it to work. Goblin Alchemist and acid ball made Warlord Swarm viable, imo (as well as other kinds).
In reference to the particular situation you mention, the way I would go about solving that would be in the following steps:
1). Make Slingers to ranged attack birds
2). Make Grunts to guard them
3). Lay down Fortified position to reduce effectiveness of birds
Not necessarily in that order, but you get the idea. Unless you buff your soldiers with zone/arena wide buffs, you will (should) lose to the beastmaster. Luckily, +2 armor for a whole zone (for 5 mana with discount) effectively increases the health of your slingers and grunts to 6 and 8 or more (66% of the time).