I'm not sure exactly how the EU Covid passport works, but any country
can theoretically pose additional testing restrictions, right?
Ok, so firstly, there is no "EU Covid passport". Each country issues its own digital certificate, and they are interoperable between the countries. But, each country sets its own rules for who can get the certificate and for how it precisely works.
Beyond that, about this question: Although the point of the digital Covid certificate is to enable freedom of movement within the EU for certificate holders, yes, each country can absolutely set additional restrictions. Here is the official language from the EU about that:
"Member States shall refrain from imposing additional travel restrictions on the holders of an EU Digital COVID Certificate, unless they are necessary and proportionate to safeguard public health. In such a case, the Member State must inform the Commission and all other Member States in a timely manner and provide reasons for such new measures." (Source)
The words 'necessary and proportionate' are doing a lot of work here. Furthermore, and important to understand, countries can also set testing restrictions for accessing spaces inside of the country, regardless of holding the pass (i.e., in some parts of Germany right now, one needs both a proof of vaccination or recovery AND a test result to do certain activities).
Or does the EU passport guarantee freedom to move about throughout the
Absolutely not. It is designed to facilitate freedom of movement, but in no way guarantees it.
If there's even a chance that we might have to test/quarantine from
country to country, I'd rather not take that chance.
Of course, it is impossible to say what May/June will look like, but if even a chance of this is a deal-breaker for you, then you may need to wait a bit longer than the first half of next year, before it is comfortable for you to travel within Europe.