Yes they are the property of the hotel, they are discretely put in a drawer for the use of guests if they wish to make use of it.
Why do you assume that? Unless the ownership is actually passed from the Gideon organisation to the hotel then I think they remain the property of the Gideon organisation.
However anything that doesn't belong to you, belongs to the hotel.
Not necessarily - if I asked whether a hotel would be prepared to display marketing materials, or even examples of luxury products, those wouldn't necessarily become the property of the hotel. They would effectively be items 'on loan' from the provider for a particular purpose.
I think it is the height of rudeness to throw away a book that belongs to the hotel and is subtly put in a drawer ( for the use of other guests) because of some prejudice some hold.
It's uneccessary.
Indeed it might be rude, but not really rude to the hotel - but rude to the Gideon organisation that provided the books. The only real issue to the hotel is that their staff are likely to need to spend the company's time 're-stocking' the bibles provided by Gideon.
Much better for them to say - 'sure this was something that hotels used to do back in the 20thC, but it isn't really appropriate in this day and age. We'd prefer not to be seen to 'side' with any particular religion so we won't be having your books in our hotel rooms.
I see no reason why the hotel cannot hold a small number of 'emergency' bibles (and korans etc) if a customer request one as they'd forgotten their own - just as hotels often will provide toothbrushes etc.
But to assume that every room should have a bible (but no other religious book) really isn't appropriate in this day and age.