The religion clause in the Constitution makes no mention of keeping religion out of the public sphere. It does not exist to protect us from religion.

It exists to allow individuals to decide, on their own, which religion (or any religion) is good for them. It is designed to prevent the government from dictating what religion or non-religion you must observe.

Sending religious-themed cards from private individuals to patients in a government hospital is not an example of the government forcing religion on those patients. The patients are still capable of deciding for themselves if they wish to accept or dismiss the religious gestures and/or messages.