Would we be happy to accept this?
Perhaps some would, but most would not.
It's not so much that they would be upset about their confirmed transactions suddenly becoming unconfirmed (or even invalid). Certainly that would happen to a number of people and those people would be unhappy. However, the more important issue here would be that the incentives built into the protocol would be proven insufficient. This lack of sufficient incentive would destroy a lot of faith in why bitcoin works, and as such would destroy the foundation on which the use of bitcoin as a currency stands.
There is a built in incentive for miners to cooperate and mine on the currently longest block. If you were secretly mining (and not broadcasting your blocks) you run a risk that the public network surpasses you and you lose all the revenue from all those blocks that you otherwise would have received. There is a large expense in mining. The revenue from the published blocks is needed to offset that loss. The protocol operates under the assumption that the risk of losing the block reward from not publishing is enough of an incentive for miners to publish their blocks. Additionally, if the secret blocks don't include the transactions, then the cabal of secret miners miss out on all the transaction fees. If they do include the transactions, then the effect on the typical bitcoin users will be much smaller since most all of the transaction will be confirmed in both chains.
I guess there are also different potential scenarios. One is the malicious version where a group hides his work. The other maybe if some miners are cut off, e.g. if the Chinese miners get temporarily cut off by the great firewall of China or some other network event. True, it is unlikely to but cut off for any length of time, but even in the malicious case, in principle, we should still take the malicious chain and build on it.
But human instinct may say no and keep the 'known' chain, in which case the underlying principle of bitcoin is compromised.
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