There is tons of great info in the thread but there is a bit context that a lot of people are missing out on.
When the PS3 had initially come out, people were still using CRT displays and dial-up internet. I would say that the PS3 really helped usher in the HDTV era with it's included Blu-ray Player. Similarly, the mid-2000s was an era of '56K WARNING' and many users had yet to make the jump to broadband internet, let alone having Wi-Fi available in the home.
The primary focus of the BC being so important though, was that the PlayStation 3 was released in that perfect twilight of old vs. new. You are generally unrestricted when attempting to hook up a PS3 to any display. It supports composite, component, SCART, and HDMI.
With 99% of PS1 games playing in 240p, 99% of PS2 games playing in 480i, and the majority of PS3 games playing in native 720p, the PS3 had to tackle the issue of all of these different video modes and had to have the ability to display them and be able to upscale them with minimal input lag. In this was the true real loss, in my opinion, of Backwards Compatibility. Without having the hardware to convert these, we've locked them behind the analog wall in televisions.
I understand that Sony removed such hardware as a cost cutting deterrent. Similarly, if Sony had remained true to their legacy of Backwards Compatibility, it may have appeared in the PlayStation 4 and kept prices up for the unit as well since it would have been an 'expected feature.' In a way, it's interesting to see now how the story developed over time. The majority of people are aware that the older PS3s do have Backwards Compatibility, but I don't believe it's widely known that it's only the PS2 compatibility that was removed. Generally when talking with people, they're unaware that every PlayStation 3 is able to play the PlayStation 1 discs.
With all of that being said, it will be interesting to see how Sony handles the backwards compatibilty of the PlayStation 5. I have a feeling it will be only for PlayStation 4 titles, with the potential to expand to any digitally released games that are currently on the PlayStation Store. I wouldn't expect any 'new' PS2 Classics to come out specifically to tout this feature for the PlayStation 5.