Eligibility for EU261 compensation?

Question for the experts here on potential eligibility under (i) EU261 and (ii) any other avenues for compensation.

I recently travelled from Innsbruck, Austria (INN) to Chicago, USA (ORD) via a scheduled short ~1hr layover in Frankfurt, Germany (FRA). My itinerary was booked through the Lufthansa website, but included flights from other Star Alliance partners including United and Air Dolomiti.

My initial flight leg - operated by Air Dolomiti (EN8051 on Jan2-25) - was delayed by ~1hr (due to inbound flight delay). As a result, I missed my connection in Frankfurt - operated by United (UA906 on Jan2-25). Representatives from Lufthansa helped me get onto the next direct flight (UA945 on Jan3-25), which did not leave FRA until 8:50am the next morning resulting in a ~20 hour delay in arrival time. In the interim, I incurred charges for transportation to and from the hotel, food, and clothing / pharmacy items as my checked bag was not released to me.

While Lufthansa representatives offered me a hotel voucher for the overnight, they noted that I was not eligible under EU261 as my flight delay was only ~1hr. This did not seem right to me as the trip booking was made under one single reservation, but hoping to get some clarity from you folks.

Happy to share any additional information as needed.

If I am eligible, what would be the best way to go about claiming this?

Thank you!!