Backwards compatibility is an interesting feature in that I actually feel that it adds more to a system on a retroactive basis than it does on when the system is first introduced to the market.
You can literally play almost every game ever released on a Nintendo handheld with just a GBA or GBA SP and a new 3DS XL that has been modded to play region free games.
The PS3 phat is an excellent system between it's ability to play three whole console generations worth of Playstation games (It's worth noting that PS1 games are running through software emulation but the PS3 stores save data on the hard drive so you don't have to worry about memory cards and I believe you can use it to play PS2 games on the Dualshock 4) along with it's extensive capabilities as a multimedia device (I've heard the PS3 has a really nice upscaler for DVDs so you could basically purchase one as a Blu-ray player and still have access to three generations worth of Playstation games.)
But the PS3 has a lot of issues that make it difficult to throw into just any entertainment system. This is what my TV setup/gaming rig looks like.
You would think I could throw the PS3 into the top shelf where the Blu-Ray player is (assuming it would fit there, the PS3 is a fairly large system) and never think about it again and considering the PS3 is such a behemoth of a device that is a very tempting option but that's not a very good idea. The PS3 phat has some issue with soldering that makes it prone to having issues with heat. Placing the PS3 in an enclosed space like that, especially one that small relative to the system itself, would only make things worse because that means any heat the system is able to generate would be trapped in the shelf with the system instead of being able to naturally move away from the system (and heat rises so the top shelf would be a really bad idea for the system.)
The PS3 phat would have to go on the top of the table so that the heat it generates can escape more freely but that presents it's own issues because the PS3 is large enough that I would have to basically hide it behind the TV. The only way I can see to get the PS3 phat in a position where I can access it's disc drive is to place it sideways in the same spot where the Wii U is. It would still be a bit awkward to put a disc in but I would be able to manage if I decided to pick one up.
But I feel like the biggest mark against the PS3 phat being used for PS2 games is that the PS2 slim exists. The PS2 slim can't play PS3 games but the PS2 slim is also the smallest home console ever released unless you count an undocked Nintendo Switch. It's so small that you probably didn't even notice that I have one sitting on the bottom shelf in that picture I posted or maybe you saw it and thought it was some sort of smudge on the glass.
You can get PS2 component cables for next to nothing and there's a really nice controller for the PS2 called the Logitech Cordless Action Controller that I highly recommend because it's just plain better than the Dualshock 2 in almost every way. The PS3 phat is a nice system but in my opinion the ease in which you can slip a PS2 slim into your gaming setup makes it easier to recommend.
I am still considering picking up a backwards compatible PS3 (Having three console generations worth of Playstation games combined with the PS3's multimedia capabilities is one hell of an offer) but I feel overall the PS2 slim is easier to recommend.