Dual-nationality traveling to US (EU + other nonwaiver country holding paper visa)

You are a dual nationality traveller, and you want to know if your EU-member-issued passport is eligible for the US Visa Waiver Program and thus applicable for an ESTA

So long as you are not a national of, or recently visited, a country on the exclusion list (Iran, Iraq, Libya, Somalia, Sudan, Syria or Yemen - there may be others), then you are eligible to apply for an ESTA on the EU-member passport, if the EU-member nation that issued the passport is eligible for the Visa Waiver Program, as not all EU-member nations are.

That is important enough to repeat: not all EU-member nations are eligible for US Visa Waiver, and thus not eligible for the ESTA program.

I have dual citizenship from both a VWP country and a non-VWP country?

If you are flying under the terms of the VWP, with your VWP passport, you must apply for ESTA. If you are using your non-VWP country's passport, you will require a visa and therefore ESTA does not apply to you.

https://help.cbp.gov/app/answers/detail/a_id/1094/kw/esta,%20dual%20citizenship/

During the ESTA application process, you will be asked about dual nationality - you should answer these questions truthfully and to the best of your knowledge.

Its also worth noting that the presence of EU-member nation passports in your group will not stop individuals being considered under their own credibility at the US border - it should not sway a border official if your child has an EU-member nation passport and you do not, when your entry is being considered.