I honestly think that one can easily achieve this by asking/PMing the user to breed the pets together themselves, and work out a price for the egg.

If I really wanted to breed with a certain pet, or a certain combo pair to breed, I'd PM the owner about it.
If I couldn't breed with them, I'd find an alternative. I'd respect the owner's wishes. Maybe find a different breeding route. It's not like it's the end of the world.

I can only see this affecting users who want to breed with older pets whose owners have left the site for an indefinite amount of time.
In this case, I cannot PM them about what I want with their pets. (That's essentially what this suggestion wants, I think -- to use someone else's pets for their own intent, if paid of course.)
In this case, it really is tough luck.

I don't really see this as necessary. All users have the ability to PM anyone about a certain request. Setting up one automatically just to breed x with y is like...

A good example is owning a pair of wheel pets, and publicizing that you're selling a breeding of the two for 400k. All the customer has to do is use the suggested feature, pay 400k, and they get an egg.
This takes away effort on the breeder's part to actually breed the pet themselves.
This, in extension, is probably one of those features that takes away a portion of "playing the game themselves."
(I link to a post by Krin about there being "a certain point where a player should be responsible for their pets". This ties into my argument that breeders should take the responsibility to breed the pets in question themselves, or the buyer should take the responsibility to ask the breeder for a specific request.)

So I do not think this is needed, if I'm reading correctly. I'd just ask the owner to breed the pets myself. If they're owned by two separate people, I would PM both owners and explain the request I'm wishing to fulfill. I've done it myself. Takes effort, but everyone owns their pets in the end, and each were notified of what I wished to do.

Interesting idea, very outside the box thinking. People tend to not have females up for breeding anyway, so I don't think the feature would overpower the site.

There could be an extra fee for doing breeding between two pets that you don't own. That way the feature is there, but going through the current methods is not lost.

I think it would work well as a supplement to the current method, rather than a replacement. c:

If this is ever implemented, God forbid, every single breeder I have will be taken down immediately.

I don't think that this is something we'd implement - not because of the reasons really talked about in this thread - but mostly because I think it takes away an element of the 'community' of the game. It's kinda like an automated /play for part of the breeding realm of the game.

As it stands now, said players can either talk to each other to breed, sell eggs, split profit or whatever they want to do. Or said player can breed, sell eggs, etc. Enabling this feature would allow pretty much just auto pilot on all of that and I think overall would diminish an important aspect of the breeding community part of the game.