I believe the best change would be to give the buzzard competitive stats for its mana cost (2/1 is weak), and change its effect to "Draw a card for the first beast you play each turn". This effect would still be very powerful, the buzzard would be more likely to survive more than one turn, and could put up a bit of a fight on its own. It also stops the 4+ card draw combos that are just too impactful. This also evens out the buzzard's chance to survive vs classes that have 1-damage hero powers and those that don't.
If the mana cost is left at 2, I would probably make the buzzard a 2/2. If the cost were changed to 3, then 3/2 is probably most appropriate, with 2/3 also being an option. This would make the buzzard somewhat similar to a Warsong Commander for its cost (although direct card comparisons across classes aren't always useful).
Buzzard is what I like to refer to as a "high variance" card, and the extreme lows and highs in its effectiveness should be reigned in (even though they want to avoid card changes now that the game is out of beta). Maybe this is a change that would wait for a new card expansion, but it should at least be considered then.