Theres actually a very simple method of modding your PS2 to play backed up games, and it doesnt require a mod chip.
Its called a pop-top hack, you simply order a replacement top to your PS2 which will be the exact same, except that theres now a PS1 like hatch on the top, which allows you to swap disks and make game backups work.
Since theres no mod chip you dont have to worry about certain games not working, or online play being an issue, you also need a PS2 swap disk, which ships with the Swap Top replacement case.
Ill get a link later,