So I did some further testing. I have a couple of SD card testing programs, although none of them write data the same way the 70D does. The card passed them, but the write speed is much lower than what I expect: between 11 and 16 MB/s depending on the test. That still should be plenty fast enough though. Read speed around 33 MB/s, far lower than the stated 48 MB/s but still OK. I haven't been able to find a program that will test using continuous writes. All of these write data in chunks, which is a different animal, but at least it was enough to verify that I got an authentic card.
The only other device I have that takes full-size SD is my older Panasonic camera, and it worked fine for recording 720p AVCHD video after a format.
While doing all this testing on the SanDisk, I ran my old but reliable Transcend card at 24/96, 4 channels until the card was full. No errors, checked the files and they are perfect. As I said, I've never had a problem with this card in the 70D.
Tried to put the new SanDisk card back into the 70D, and it wouldn't let me do anything with it now: I got a "bad card" message or something to that effect, it told me I had to replace the card, and it powered down. This was right after doing the video recording test.
Put it back into PC, used the official SD Formatter utility to do a full/erase format. Back to the 70D, and it formatted without incident and now seems to be working fine. I just recorded a couple hours of 24/96, 4-channel and no errors, popped the card into the PC and verified that there are no glitches on recording either.
It's worth noting that after I had all of my problems last night I tried formatting in the 70D using both the Quick and Erase methods, and continued to get Write Timeout errors after 10 seconds. That continued today, and the only thing that fixed it was formatting the card on my computer using the SD Formatter utility.
I'm not sure what's going on here. Did the filesystem on my SanDisk card somehow get so corrupted that the on-board format routines couldn't fix it? And why was it not even recognized in the 70D after formatting in my camera? This card appears to be working fine now, but will that last? All I can do right now is continue the same test I did before until the card is full, and see if I get any problems or this was a one-time thing.
Maybe Tonedeaf and any others who have purchased this same card would please run a continuous test also to see if this model of card really is no good with the 70D and TEAC Japan wasn't thorough with their testing, or if it's just mine. Card model is SDSDUP-16G.
EDIT: I'm running that card in the 70D now again until the card is full to see if there are any further problems. If I can do that 2 or 3 times with no issues, then maybe this was a fluke. This card is going to have to prove itself worthy before I use it again to record something I really care about though.
One other thing I forgot to mention before: for the marching band show I was recording last night, the set that I cared about for our students actually came out fine. What I do with these recordings is I replace the audio on the video track from my phone or from another camera using Vegas, and then that goes to a private share to use for the students to self-assess following each competition. What I noticed this time was that even though this particular 10-minute recording was free of audible glitches, it drifted significantly from the video track such that I had to do a cut/crossfade to make the remainder match up. Even without timecode sync, I never have significant drift for recordings this short - in fact it's the only time I've ever noticed a mismatch at all. So a longer-than-normal recording may be another symptom of card write problems.