If I wait until 5 days before my flight how likely is it that I will be able to get both an appointment and receive my emergency passport in those 5 days?
Very likely. The embassy wouldn’t be imposing this restriction if it meant that substantial numbers of citizens were unable to fulfill their need for an emergency passport.
it would be possible to apply for an ESTA and enter the US using my Irish passport, then renew my American passport when I arrive to then be employed as a US national and leave the country on my American passport?
You can certainly apply. Your application may be refused. Even if it is not refused, the airline may notice that you are a US citizen and refuse to allow you to board the plane. If you are flying through a preclearance airport (from Ireland or Canada to the US, for example), the US immigration inspector may prevent you from boarding the plane. In short, there’s a good chance that this wouldn’t work.
If you can get into the US using your Irish passport, that doesn’t change your employability. You would nonetheless be a US citizen and therefore allowed to work even if you don’t have a US passport (though you will need to fulfill the requirements of the I-9 form, and the simplest way to do that is probably with the US passport).
I cannot apply for any work visas as the company made explicit I was only being hired as a US national and they cannot assist with visa applications.
Anyway, the US (like every country I’m aware of) does not grant visas to its own citizens, so you would never get a work visa. This brings up another point: if you do manage to board the plane with your Irish passport and associated ESTA, you still ought to present yourself at the border as a US citizen who doesn’t have a passport. You’ll get a talking to, perhaps, and then you’ll be admitted as a US citizen.
But again, even if somehow you enter on the visa waiver program, without mentioning your US citizenship, with a stamp in your passport saying “WT” (“waiver, tourist”), that doesn’t change the fact that you are legally a US citizen, entitled to work, and entitled to remain in the US indefinitely.