No.
You see, PS2's CAN run burned cd's and dvd's. They're just programmed NOT to. When you put one in the PS2 and start it up, it recognizes that the disk is home-made, and it's like "NOPE I WONT PLAY THAT."
What Swap Magic does, is gets you past that burned cd/dvd check that the PS2 does, and then STOPS spinning itself, so you can take it out of the disc tray and put in the burned cd/dvd, press X, and then start playing your dvd-r or cd-r burned game.
HOWEVER, there are fail-safes in the PS2 against this. First of all, the eject button cuts power to the PS2, so you can't just press eject and swap disc, it will reset the PS2. There are also one or two other mechanical sensors that trigger when the tray is opened that do the same thing. If any of these things are tripped, the PS2 reset's, and you have to start over, so you see it prevents you from swapping out discs.
UNLESS you buy and install a flip top. It's simply a replacement for the tray cover on the PS2 (SLIM), but it has a hole in the top (with a cover that flips down). It allows you to switch out discs without tripping the PS2's sensors. You have to disassemble your PS2 (and void your warranty) to install it, but there's guides online, and it isn't TOO difficult. But you'll need some precision screwdrivers (ask for exactly that at a hardware store).
Total cost: ~$40.
That's for the PS2 Slim. The fat PS2 requires a bit more work, I hear.
edit: This also allows you to play imported games.