The Eurofighter could of course carry and drop US nukes. The issue is that it would need US cwrtification to do so, and for rather obvious reasons that certification was close to impossible to get.
Again, the F-35 is a great plane. There are reasons so why it is not not even for the USN and USAF, only one.
Edit: The reason for the outcry over French nukes is not that they are French, but rather the need to change the laws around nuclear participation to allow the switch. And that legislative changes would lead to a ton of public debate everyone wants to avoid, as this debate would kill nuclear participation in general. So nobody in governmant wants to take this risk, hence the procurement of a US platform to replace the aging Tornado fleet as a carrier of US nukes.
And the French, which excluded their nukes from NATO oversight, might actually be reluctant to share them with Germany. Using Rafales insteads of Eurofoghters might helo, but then Germany is in the same position it is with the F-35. With the added "benefit" of a potentially hurtful public debate.
But German defense policy is built around self-defense and coalition defense: NATO, EU, UN. Nuclear weapons are opposed by Germany, and it's only because NATO is viewed as essential that nuclear sharing is accepted. Most Germans don't know about it though. A bilateral (FR-GE) extension of nuclear policy would be politically unacceptable to Germany, and require extensive legal change.
I think the German acquisition of F-35s is a bit subtle. The Luftwaffe always wanted to buy F-35. I suspect they stalled the subject of the Eurofighter nuclear certification (which would take a good ten years) to force the German government to buy F-35s to keep its commitment to nuclear sharing. The whole thing happened at a time where the ministry of defence seemed disorganized and plagued by scandals. To be coherent with the Eurofighter policy, nuclear certification should have been a top priority ten years ago...
The relationship between the German government, the Luftwaffe, the US industry (and Lockheed...) and the German aerospatial industry has always been complex, and Germany has often been keen to use allied programs to avoid internal defense industry problems, with more or less success. See the Starfighter article on Wikipedia for the most egregious example.
With the German industry stalling the French-German-Spanish fighter project, I also suspect that they'll plan a similar manoeuver with the Eurofighter replacement.
The main pain point in all those large European defence programs is industrial workshare, especially between France and Germany and within Airbus between various sites.
You need an extensive set of avionics to control and arm the nuke in the plane which add substantial cost and aren't designed for/integrated into all aircraft.
I presume that they require US launch codes (and the F35 has a kill switch?). So what is the use for Germany?
The fear in Europe (confirmed by Westpoint graduate Mearsheimer long before the Ukraine war) is that if Russia strikes a single city in Europe, the US will get second thoughts and not retaliate. Germany then cannot retaliate either, so what is the point?
This is the reason for France to have its own nuclear arsenal and fighters.
Which begs the question, why didn't Germany collaborate with France on nukes where they could be a more or less equal partner, compared to the US where they would always be the subservient party with little to no control?
De Gaulle wanted sovereign control of any French nuclear deterrent and did not trust that, if it came to it, the US would put itself on the line for Europe: https://www.nato.int/cps/en/natohq/opinions_139272.htm - presumably, for similar reasons he did not want any dependence on allies, even ones closer in location and likely part of the same conflict.
The US did actually help the French nuclear programme once it started going, under the reasoning that multiple nuclear-armed NATO members is better than one.