Am I somehow exempt from ETA and EES?
Of course you are.
Systems such as ETA and ETIAS (not EES) are designed to serve as a lightweight pre-screening for visa-exempt travelers -- lightweight in that it is not as thorough as the screening of a visa applicant. You are not traveling with a visa exemption when you travel to the UK because you have ILR. You are not traveling with a visa exemption when you travel to the Schengen area because you are an EU citizen or a citizen of a non-EU Schengen country. There's no need to qualify your citizenship with "technically"; you are a European citizen, plain and simple.
There are no cracks to fall through here; in neither instance do you belong to any group of travelers who are required to apply.
In a comment, you wrote
I am not so much worried about being tracked. This question is more about being refused entry by border controls. Going either way. Having a passport alone is no longer sufficient.
In your situation, for the Schengen area, that statement is incorrect: as a citizen of a Schengen country, a passport alone most certainly is sufficient. For the UK, the statement arises from an incorrect assumption: you aren't traveling to the UK with a passport alone but rather with a passport and ILR. If you don't have a physical document showing your ILR, you still have a record in some database somewhere that shows you have ILR, similar to ETA, which is also primarily a record in a database.
Note that the information at the second link in the question applies "If you are travelling to a country in the Schengen area using a UK passport," so it does not apply to you. I suppose you realize this, since you offer it in support of the proposition that you are not subject to EES, but I think it bears explicit mention. A more authoritative source in support of that proposition, which furthermore addresses it directly, is https://travel-europe.europa.eu/ees/whom-does-ees-not-apply_en:
The EES does not apply to:
- Nationals of the European countries using the EES, as well as Cyprus and Ireland
- ...
For ETIAS, https://travel-europe.europa.eu/etias/who-should-apply_en:
Who does not need an ETIAS travel authorisation
You will not need an ETIAS travel authorisation if you are:
- A national of a European country requiring ETIAS
- ...