Even assuming a best case scenario where nothing goes wrong the Saturn V and especially the LM aren't going to be available much sooner than OTL. You'll get two or three manned orbital missions, all very similar to Apollo 7 flown in 1967, but whenever the Block II module actually becomes available you'll still need to have an orbital test. Remember that the only real reason Apollo 8 didn't carry an LM was that the vehicle just wasn't ready yet. The early flights on Block I probably don't really amount to a whole lot, and I rather suspect that if successful no more than two would be flown. Such a TL might be able to push Apollo 8 a few months earlier but other pieces are going to hold up the landing as OTL.
There could be some butterflies around Apollo Applications and Skylab of having flown the Block I I guess, though not many of them are positive. More Saturn IB's will have been expended, more funds spent and more time spent in LEO. If anything this would end up reducing the time spent on Skylab.
All in all, I don't see avoiding Apollo I changing much if there isn't another accident, and overall it's seems a lot more likely that if not Grissom, White and Chaffee something will happen to someone else. Just about any other likely accident scenario I can imagine is likely to cause more of a delay in fact. Just the failure coming later will throw the program into disarray, but if it happens to a block II capsule, or on a rushed LM flight I don't see much hope of recovering to any schedule that lands in 69. IMO the only way to land sooner than OTL with a recognizable Apollo program is to make the LM program go quicker and smoother, which frankly borders on ASB given what they were doing.
PS: As far as Grissom making the first landing I think it really is pretty likely. Slayton aside the whole program pretty clearly wanted to land one of the original seven, and Grissom was always the obvious choice. Depending on time lines and crew rotation he might have ended up on the second landing, but really he was arguable the most experienced commander available (I tend to think that;s not really fair to the guys who flew twice on Gemini and as such had more practical experience, but so it goes), and there would be a lot to be said for that on ATL's Apollo 11 which was even more so than the other landings nothing more than an engineering test. Certainly he would have got one of the landings barring something happening to ground him.
