First of all, keep in mind that on the 360, saved games and Xbox Live profiles are ultimately stored in different "places". While you can keep both on a hard drive (or on a memory unit or flash drive if the saved games are small enough), saved games are only kept locally, while your Xbox Live profile is tracked on Microsoft's servers.
If you're talking about an Xbox Live profile, then it will "move" the first time you recover it - once you recover it from Microsoft's servers, that becomes your up-to-date profile, so you won't need to copy it from your hard drive. Your saved games won't be accessible until you get that enclosure so you can connect your old hard drive.
Local profiles, ones not associated with an Xbox Live account, are only stored locally, so if that's what you have, then you'll need to wait for the enclosure to arrive.
Once the hard drive is connected to your new 360, you can move whatever saved files will fit to the internal flash drive. (If you go to My Xbox, System Settings on the far right, and choose Memory, it'll show up as Memory Unit (3.x GB free), and you can copy files from your hard drive to it and vice versa.) I don't personally believe there's a difference between storing your profile on the internal drive and storing it on the hard drive. However, if you find yourself playing away from home on occasion, you may be better off picking up a small USB flash drive (up to 16 GB), formatting it for the 360, and storing your profile on that. You can simply pull the USB drive and take it to your friend's house without needing to recover your profile at either place.
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.
Best Answer
Destiny lists itself as "requiring an Xbox hard drive" because it requires an internal Xbox hard drive. In this particular case, it turned out that your console was the "arcade" variant, which does not come with an internal drive.
Please keep in mind that future releases will eat up more hard drive space.
Along with the core Destiny game, there are currently three DLCs1;
The Taken King can be considered more of an expansion, and brought a lot of changes to Destiny. These changes effected the vanilla version, too.
Returning players may find that the same occurrence happens with an update, as game updates are frequent, and can often add or change content.
Players that have an account registered prior to September 15, 2015, may be entitle to a free memory upgrade.
By contacting Bungie through their customer support, players may be entitle to receiving a free USB designed for their Xbox 360 or PlayStation 3, with the latest updates from The Taken King.
1At least, as of 6/4/2016.