Question: Do I need a EU passport or EU ID card to legally work in the EU (or establish that I have the right to work in the EU)? Or is a certificate of citizenship sufficient?
Legally, your right to work is not contingent on this and there is no Europe-wide rule that makes holding any document mandatory. Importantly, if you do start working anyway, you are not committing a crime and cannot possibly be banned or forced to leave the country. You do have the right to work from the day you became an EU citizen and if any doubt arises down the line, you should be able to clear it up later.
In practice, employers are sometimes supposed to check you are allowed to work (and for that would require some proof of your citizenship) but they don't necessarily need a passport or ID. What's typical on the other hand is that you have to provide an official proof of address (in the countries where you have to register your address with the authorities) and the local social security, insurance, or national tax number. Both of these will require dealing with the authorities and will be considerably more difficult, if not downright impossible, without a national ID card or passport (in fact it can even be difficult with a passport).
I worked in multiple EU countries and I don't recall always having to present my ID to employers. I recall at least one instance (in Germany) where I could start working without one (it had just been stolen) and another one (in the Netherlands) where I started on the day after I arrived, without official address nor tax number (BSN). In both cases, I was expected to solve these issues within the first month and you risk a fine if you don't register within a week or two but it was neither illegal nor impossible to start working before all the formalities were completed.
None of this means I would be completely comfortable about being months without a passport. But the main issue for you will be entering the country and what your employer's HR department is prepared to tolerate, not any sort of legal obligation to hold a passport to work.
Note that in one of the cases I described above I went to the local consulate to get an emergency passport. It wouldn't have been possible back in my country of citizenship but there are some special procedures when you reside abroad. These rules change all the time and depend on your country of citizenship but that could be worth a try.