Stable is not a thing in natural populations
Let's consider herd animals to start with.
If a male is successful, that male will have lots of offspring every year, or perhaps for only one year. It's a high risk strategy, an unsuccessful male might have no offspring ever. A female might have a maximum of one offspring each year, but will have that offspring every year.
Female looks like a safe bet, right?
Not so, if there are a lot of females then there's a vast opening for more males, and hence males will be more successful. But now there are lots of males, so the reliability of a female's regular breeding carries the flag.
This creates an on going oscillation in the populations of the two sexes, but at no point is the situation stable.
So where does the advantage come for each of your sexes?
The strongest is Male/Egglayer/Brooder. This obviously fills all slots, will breed every season but could breed multiple times with other egglayers or egglayer/brooders.
Next is egglayer/brooders, these will breed successfully every season at the baseline of the herd female.
The male/brooder is an interesting one, there are two examples in nature, seahorses and ostriches. It's clearly a valid strategy, they're likely going to attract egglayers, though other egglayer combinations are going to seek them out for a bit of added seasonal insurance.
The female/brooder is the classic female bird (or herd animal) we're all aware of. I put this below male/brooder because laying eggs requires more resource investment than producing sperm.
Male and egglayer are at a disadvantage, they either need to find a combination of {other}/brooder or both an {other} and a brooder. The egglayer also suffers the higher investment of laying eggs. These are the equivalent of the herd male, high risk strategy but could be really successful as they don't have to invest in actual brooding or raising of offspring.
Brooder. A genetic dead end in its own right. There would need to be a very strong family group system for them to be anything other than a burden on the parents to raise.
But what if they're small birds rather than herd animals?
Now you get the advantage of the brooder being a third parent. If you can get one that is. They're still going to be the neglected child if there's any risk of one starving during them being raised. Brooders aren't going to be common.