The Xbox 360 has whats called a "DVD key" paired to each motherboard and DVD drive combo. The DVD key on the motherboard must match the DVD key on the drive or the console will not boot games. If you already tried dumping the DVD key from the drive that came with the Xbox and flashing it to your replacement drive (using the correct firmware for the replacement drive as well) and the console still does not boot games (displays "Play DVD" when a game is inserted), you don't have the correct DVD key and the console will never boot a game.
There are ways to retrieve DVD keys from the motherboard however they fall under "ridiculous hardware hacks" as you called them and will probably be more effort than it's really worth. However, if it's a route you're interested in following, there are currently two methods for dumping the NAND (Where the DVD key is stored) of an Xbox.
One method is the JTAG hack (tutorial) you would only need to follow this tutorial up to the point where you dump your NAND then you can use another program to read the DVD key from the NAND. However know that the JTAG method only works for Xbox consoles running kernel version 7371 or lower. (You can check what kernel your Xbox is running under the system info.)
If your kernel is above 7371 though you'll need to do the Reset Glitch Hack or (RGH for short, tutorial). I don't know much about this hack because it's fairly new and I myself have been out of the Xbox modding scene for a while now. It's going to follow a similar style to the JTAG hack though you'll have to solder connections to various points boot Xellous and dump the NAND.
Those, unfortunately, are the only two ways to dump a DVD key from a motherboard, so if you don't feel like going through the effort of the methods posted above, your Xbox will not boot anything besides DVD movies.
While we're on the subject of DVD drive flashing and replacement I'll clarify some things for you for future reference.
Some places say you can just take any old drive from another Xbox 360, but that didn't work - I still get the same error.
You can indeed use any model of DVD drive found in the Xbox 360's as long as the DVD key on the drive matches the DVD key on the motherboard; if the keys don't match, your console won't boot games. However, if you intend to use this console on Xbox Live, you must replace the drive with the same model drive that was originally in the Xbox — if you don't, you will be console-banned from Xbox Live. (Meaning that console will no longer be allowed on Live.)
Some places say you need to have the exact same manufacturer and model DVD drive, and it should work. I have reason to suspect this is not the original drive (see below), and there is no place which can tell me what model DVD-drive my model Xbox should have come with.
Like I said above, if you wish to just play games and not access Xbox Live, any DVD drive will work so long as the DVD keys match between the motherboard/drive. If you wish to access Xbox Live, your best bet of keeping your console from getting banned is to use the model of DVD drive that originally came with that console.
Here is the list of Xbox 360 motherboards. Generally the motherboard DVD drive pair goes something like this.
- Xenon — Hitachi, Toshiba, or Philips model drive
- Zephyr — Hitachi or Toshiba model drive
- Falcon — Benq or Liton model drive
Every revision post-Falcon contains some variation of a Lite-on Drive for security and performance reasons.
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I've regularly been using my Xbox Live account on 3 different consoles. If I remember correctly I didn't use a USB drive, instead I just retrieved the account from the magic of the internet. I'm not sure about what @TZHX means when they say that you'll have to re-download your account every time you switch, if he is right about that it's not something that you actually have to do, as the Xbox will do it for you. The only thing I've noticed is that when I sign on to a different console, a message pops up that says "This profile was last used on a different console."
The only downside that I can think of at the moment is that if you get achievements while offline on one console and go online with the second, those achievements may be lost. So all in all yes, you are correct it is possible and the Xbox makes it really easy for you.