One suggestion that I haven't seen mentioned yet is to enter Europe through a country with an established, consistent process for issuing a European proof of vaccination QR code to a person visiting from abroad.
France is such a country. Formerly, you could apply online for free before traveling, and upload your ticket and your CDC card. Now, you must go to a pharmacie and pay them to read your CDC card. At least the process is standard and the fee is capped.
Once you have a QR code from any EU member country or from cooperating neighbor countries like Switzerland, you can present it in any of those countries. Showing your French QR code on paper in Germany is fine. Showing it in the French Tous AntiCovid app, in Germany, is fine. Importing it (from paper, or from the screen of a second device) to the German CovPass app is fine. All options work. It is, fundamentally, a Europe-wide standard QR code.
I imported my French QR code (obtained through the original, online process) into CovPass, Luca, and also the Swiss Covid-Cert app, with no problem. If importing from the screen of a second device, be sure to choose «Partager» first, in the Tous AntiCovid app. This allows you to save a working screen shot, to send to your other device. Display that image on the screen of your other device and scan with the camera of your main device, to import into the national app of your choice.
Be aware that Schengen-internal travel restrictions can be reimposed at any time. For example, France might drop off another European country's green list, requiring you to register electronically before entering the new country from France, or to get a test before or after entry, or even to quarantine upon entry.
If you are planning a multi-country trip, it does make sense to choose your starting point based on the ease of obtaining a European proof of vaccination QR code.
(Note: As a handwritten paper document not backed by any national database, the CDC card can never be verified — not even in the US. It can just be read and trusted. US states are all required to have vaccination registries and could conceivably verify the data, but US states and foreign health authorities don't talk with each other. The QR codes issued by some US states, like California, follow a global standard, and the fact that they are digitally signed, i.e., that they haven't been tampered with, could he verified anywhere in the world, but unfortunately, the EU and its neighbors have chosen a Europe-only standard for their QR codes, and won't pay any attention to QR codes from the US states that issue them.)