They lack the power to take on something so big and tough.
They were getting winded and worn out against Chitauri Leviathans, which are much more fragile than Kaiju. Kaiju can just shrug most of their attacks. Even Thor's full power lightning is likely to just cause a burn, but not a kill.
And that's putting aside dealing with, say, Leatherback, who's EMP may shut down Iron Man (/Iron Patriot) right out.
Any of the three "main" Superman versions (Reeves, Routh, Cavill) would easily end the Kaiju threat in five seconds flat.
Manhattan could.
Silver Surfer could.
Frankly I have my doubts that the Avengers are as boned as you all say.
The aliens didn't just shit out the entire Kaiju army in one go. It was a process that took years.
Are the Kaiju really big, durable, and strong? Sure. But they are also slow. Thor, Iron Man, and even Hulk would run circles around them, and though they lack the raw power of the Jaegers for the most part (Only Thor can really contest this claim by feats), they are so much smaller, more maneuverable, faster, and can apply concentrated force that the relatively huge and clumsy Jaegers couldn't match. I would compare them to, in this context, the Colossal Titan from Shingeki no Kyojin/Attack on Titan. Easily the largest, and most powerful Titan seen thus far, and capable of destruction no other Titan can begin to match. But so damn slow and clumsy that he is far less of a threat to trained soldiers in 3DM gear than smaller, weaker Titans. Maybe the aliens could breed much smaller, quicker Kaiju that could fight the Avengers on better terms, but that is speculation.
If all the Kaiju come out at once? Sure. They're screwed. If it is at the same rate as in the film itself? I don't find an argument that the Avengers could win too hard to believe.