Since last year when a console is replaced Microsoft updates the DRM to the new serial number so that games still work without being connected to Xbox Live. I have a few friends that have received new Xbox 360 consoles with new serial numbers and their content is fine. I have no doubt that some people may have had issues if someone did not properly update the DRM to the new serial number, then they would be talking to many people from customer service to get it fixed.
I have had one launch day Xbox 360 fail back around June or July, and I did have the problem with content not working. Support did decide to reimburse all my points, however I got tired of the support person I was talking to (first person was good, this one was not) and asked to talk to someone higher up. This person knew exactly was he was doing and said it should have worked, and hen asked for the serial number and one other number (it's on the blade section where the serial number is). The next day when I connected back into Xbox Live my console was updated so all my content now worked. I was happy and did not need to points anymore, however they still gave me the points anyways "for my trouble". One of my friends console also failed a week before mine and another about a month later and they both received new consoles and all their content works for anyone on the system without being connect to Xbox Live. Another failed about 2 months ago and his is fine too.
If someone does have an Xbox 360 that was replaced and it does not allow access to the content they purchase without being online, you need to get a DRM transfer to the new console. If support does not know what you are talking about you need to talk to someone higher up then that person. The people that answer the phone are first tier and typically only handle the simple and routine problems. This is that way that every large corporation work and is certainly not unique to Microsoft.