I'm a little late to this discussion, so I hope what I add will be of value to someone.
The problems you're having is why I use a product called Scannerz. There's a lot that can go wrong with a system, and it can't always be traced properly by playing a "guessing game" or swapping out component by component. Simple surface scans just won't cut it, especially if all they look for are I/O errors (which is all most do.)
First, because I personally think the Scannerz documentation is somewhat confusing, allow me to explain it to you in a more "real world" example.
Suppose you have an old LP (vinyl record) or a music CD-ROM that you play on a stereo system. If, every time you start playing a song on the media, and at 3min 21 sec into the song, if it's a vinyl record it skips, or if it's a CD it jumps around and malfunctions, then the problem is repeatable. It occurs every time you play it, and it occurs at the exact same spot in the song, so you're pretty sure (about 100% in fact) that the media is bad. The solution in this case is simple: replace the media.
What if, instead of having a repeatable problem, it's erratic. Maybe the sound drops out with a pop on one speaker intermittently, maybe the sound comes on and off erratically, maybe there's lots of static or hum, but it apparently can't be traced to the media per se. What do you do? You start checking connections, grounds, cables, etc. and you troubleshoot it. Maybe it's a failure in the receiver itself.
Problems with a computer system are somewhat similar. There's a lot that can go wrong with them. You could have a bad hard drive, a faulty drive cable, an intermittent fault in the supply connections to the logic board, but you can't find this out if the problem isn't isolated properly.
A computer system differs from an audio system in that the audio components typically just throw their output to the next device without caring about whether or not the signal is received properly. Computers are different. They go through a process of handshaking before the events can proceed properly, and if the handshaking doesn't occur properly, the event may fail.
Although many people think Scannerz is hard drive scanning software, it's fault detection software - it uses the progress of the scan on a hard drive as a reference point. Just as someone listening to a faulty recording media can determine whether or not the media is the problem because it occurs at the exact same spot, Scannerz allows you to determine whether the problems are related to the drive or somewhere else in the system because surface scan problems are repeatable, system problems are not.
Scannerz will detect a disk problem at the exact same spot if the surface of the drive has problems. However, if the problem is elsewhere, for example if the cable connecting the drive to the system has intermittent faults in it the problems, just like a speaker with intermittent connections in it will produce erratic and unrepeatable results. This is why Scannerz ability to use a cursory mode (this is Scannerz, not Scannerz-Lite which doesn't have these capabilities) is invaluable. If you run a test on a drive and it reports failures, you can re-scan them in cursory mode and if the problems repeat, you know the drive surface has problems. If they don't repeat, then you know the problem is elsewhere. Scannerz monitors system timing, and if problems are present, they'll be exposed either as errors or irregularities (the time it took to complete an event was out of bounds.)
You can isolate the problem by using path isolation. In your case, they've already replaced the logic board, so, unless you're extraordinarily unlucky, that shouldn't be the source of the problems. The remaining sources are likely the hard drive, the cable connecting the hard drive to the system, the supply and/or connections to the supply/battery, or the surfaces of the hard drive (i.e. your suspicion about the hard drive being bad are verified).
You can start isolating this by booting from an external FireWire or USB bootable media, unmount the internal hard drive using disk utility, and see if the system functions properly. If it works great using only the external media, then either the hard drive needs checking or the cable between the logic board and the internal HD is bad. If this is the case you'll need to get the software to test it (or make Apple test it for you, after all, they supposedly fixed it).
If the problems persist even using an external drive, it indicates that the problem isn't the internal drive or cable related, which pretty much points at the supply and/or it's cables (if there are any). Odds are when they swapped out your logic board, the didn't replace the cables, and if they were defective, the problems would persist, as they would if the drive's surface platters are defective. The test software Apple uses is simple surface scanning software and it won't necessarily be able to catch or verify intermittent problems.
I hope this post is of value to someone.