I see how you can get that impression from the outside. But for the first three Merkel administrations the CDU/CSU very firmly ruled out any coalition with the Greens, which were somewhat left-ish at the time. In that timeframe they undid most of the energy policies ("Ausstieg aus dem Atomausstieg") enacted by the SPD-Greens coalition of 1998-2005, and then later reversed parts of that because public support for nuclear energy collapsed even in the center-right part (CDU, CSU, FDP) of the political spectrum in 2011 due to Fukushima. Some call it "Ausstieg aus dem Ausstieg aus dem Atomausstieg". It's far more expensive than the original nuclear exit plan - pretty much a golden parachute for nuclear plant operators, and also caused a significant ramp-up in coal.

You're right though that before the final Merkel administration the Greens, the CDU/CSU and the FDP had talks, which ultimately failed because the FDP passed the CSU (the right-wing regional appendix to the CDU) on the right, which was a bit too much for the Greens, which at the time largely occupied the same center-ish position they hold today, having shifted right considerably in the time between 2005 and 2017.

Overall the impact of the Greens party has been very small on the Merkel administrations, however, some parts of the green movement (which is not closely tied to the Greens party) in Germany are the majority opinion in Germany (especially when it comes to a heavy anti nuclear stance) and that's sufficient to drive Merkel policy making. It's also important to note here that Germany has a very strong NIMBY attitude, so while the majority "wants" green energy, they don't want any of the infrastructure in sight.

I'd also like to point out a fundamental difference between Bundestag culture and US Capitol culture. There are bi-partisan bills in the Capitol, but in the Bundestag it is customary for the ruling parties to unconditionally veto drafts introduced by other parties - exceptions are rare. Almost all votes in the Bundestag are under Fraktionszwang i.e. MPs deviating from the party line won't get on the election list next time around, so don't have a safe seat any more (because parties in Germany can't literally force their MPs to vote their way, they can only "incentivize" it).