I don't doubt the club vs country logic, in terms of potential quality of side.
But it depends on how you look at biggest stage though. The Champions League final has yet to reach 500 million viewers Globally, whereas the World Cup final has anything from 1-3 billion viewers, depending on where you read it.
The World Cup gets far more viewers than anything else, as it attracts so many casuals, and a good World Cup run can capture the nation. Look at the scenes in France and Croatia alone yesterday. So in terms of 'eyes on', it's without doubt the biggest stage of all. The fact it's every 4 years as well, when club competitions are annual, makes it even more of an occasion to perform in, because you won't get many chances at it. Hence why it matters so much for a player to perform well in it. It carries far more pressure and reward than any other footballing competition.