The EPP could have put its foot down and said "Nope, we'll only vote for Weber"... which probably would have deadlocked the election and caused a constitutional crisis.
And really this is what the Spitzenkandidat system is, always was: It's the parties saying "we field this guy or gal for Commission President" and the EPP did try its best to push Weber but ultimately failed. It never meant "The Commission President is now elected directly by the people".
In any case the political power play is within the Parliament, not within the Council. The Council can come in and suggest a candidate if the Parliament is hung but their role has shifted from actually deciding to being a mediator, "hey wouldn't this candidate be a compromise that can be reached". In practice, that means, also going into the future, that if the parliament gets its act together and agree on one of their candidates, that's going to be the candidate, if it doesn't, the Council is going to hum and haw until they've found a politician from the party which won a relative majority that is acceptable to the other parties.
It's not precisely the German system where parliament alone proposes and elects the Chancellor so maybe the name is a bit off but it's similar to other European countries where the president or monarch suggests a candidate to parliament. (In Germany, all the President does is swear the Chancellor in, in their role as notary of the state.) That kind of position doesn't exist in the EU, so the Council it is. Arguably more democratic than, say, Spain, as the Council does have democratic legitimacy while monarchs don't. It's like if the Congress of Deputies proposed a candidate instead of the monarch. Though of course the Council is not made up like the Congress, like the Bundesrat it isn't made up of deputies elected specifically for that purpose, but state governments.