I know this is an old thread, but I had this exact issue.
Yes, down deep in the light engine, there is a small plastic lens. Apparently, the newer bulbs burn hotter than the original. And so melt this lens.
This is what is causing the black shadow centered on the screen. And it will only get worse.
The problem is...even if you can find a replacement lens and put it in, it will do exactly the same thing. Same plastic construction.
I managed to salvage mine by splicing in a glass lens. I carefully cut out the melted part, and found a small magnifying lens of about the same size. Glued this new lens in, and it has worked for several years. The only problem is that the new lens is a degree or two off, and so the picture is slightly skewed onscreen.
But it was a free TV, so I had little to lose.
At this stage of the game I'd just get a new set. New ones that size are cheap enough to not bother trying to fix the SP61MD10.