I agree partially--grades matter somewhat.
However, if you have all Ps you'll only "likely strike out" if you're not strategic about how you bid (ie: only bidding Bay Area, only bidding top firms, etc). Stanford is good enough that even with straight Ps ITE your chance of getting a market job is solid if you're not picky about where that job is or who that job is with. Think about it this way: coming out of Stanford 1L with all Ps, you'll have a choice between getting a market paying job somewhere or getting some job--potentially paying less--in the location of your choice.
Obviously, these things are influenced by other factors--your pre-law experience, your summer job, what journals/activities you did in 1L, how you interview, etc. If everything else on your application is working against you, I guess all Ps could be the final straw, but if you have a lot of other positives in your application (as most people do), the grades probably won't be a huge problem. Also, it's important to note that things will likely be marginally better next year than they were this year (and things seem to have been marginally better this year than they were last), so all of this advice is probably on the pessimistic side.
Anecdotally, I can say that many of the less competitive firms I interviewed with didn't look at my transcript and were clearly only interviewing for fit (for better and for worse). I also don't know of anyone who struck out during OCI (although I imagine there are a couple), and I know of several people with all Ps who have at least one offer.
My sense is that Stanford's relatively better positioned than Harvard ITE, and somewhat similar to Yale; there are more firms that want Stanford students than there are Stanford students to go around both because of the small size of SLS and the significant geographic spreading of SLS students (there are obviously relatively tons of SLS students who go to LA/Bay Area, leaving relatively fewer for NY/DC, and almost nobody for Boston/Chicago/Texas). Firms DO care about school diversity in their classes, at least when it comes to HYS. The one exception is if you want to go to a boutique in Boston (then Harvard/Yale might be better) or Chicago (then U of C or Northwestern might be better). Then again, you'd be in much better shape going to Stanford if you want to go to a boutique in the Bay Area, or likely LA. Most boutiques only take 1-2 students every year, and they generally already have their established "regional" pipelines.