"I really thought in 1904 that I had a solo shot. I am 100% sure that I timed some stabs wrong, and messed up a guess in losing Kiel. Where did I go wrong?"
I will freely admit that getting to 18 centers is my weakest point. If you want to make it to a draw, I'm your man, but I'll see what pointers I can give. If you want to solo, you need to start counting your centers early; the earlier the better. If you don't, you'll find yourself stuck at 12-17 centers with nowhere left to go. With that in mind, let's start looking at the game:
1901:
Right from the start I have criticism. Why bounce England in NWY? If you want to solo, you'll probably need Northern England. Why race France to it when you could use England to stall for you? Let him fight France while you focus in the South and then gut him from behind when you are ready to assault him with full force.
1902:
I liked the Spring of 1902, my opinion of Scandinavia notwithstanding. I'm surprised you turned on Austria in the Fall when he was so eager to help you take Turkey. I'm sure BUL was tempting, but I would have taken the help to quickly take out Turkey. By artificially propping up Austria, you could have also quelled fears about your rapid expansion.
Meanwhile, you'll note that England and France started working together against Germany, partly, I suspect, because he knew his north was not secure. As Russia you want a weak France, preferably England and Germany over-committed in the West. You certainly don't want England working with France.
1903:
1903 gets a bit odd. Your re-stab Turkey, un-stab Austria, stab Germany, and un-stab England. Very busy. The problem now is that you have the trust of no one and the only people you are helping (France and Italy) are the exact two countries you need weak to solo.
1904:
This is where your mistakes came home to roost, imho. You say in 1904 you thought you had a real chance to solo. I say by 1904 you had lost your chance. England collapsed and you had none of it. It's true you finally got into Turkey, but you got less of it than you would have with Austria. Instead of Austria overcommitted with an Italian war and a nice backside for you to take, you have a strong Italy and Austria Proper full of armies.
1905:
1905 was a pretty good year. You seem to have realized you needed Austria and you made good progress in the north. By this point, though, I think France had gotten too big. Let's see:
Russia+Turkey+Balkans+Scandinavia+Berlin is 13. Where can those other 5 centers come from? You could take the rest of Austria, but it's unlikely you'd get GRE or TRIE. You certainly weren't going to get KIE. Maybe EDI, but that still only gets you to 17.
Now it's important to note that France is getting very strong, too. With that fleet build in MAR, you may have been able to convince Italy that France was the solo threat, not you. The problem is by this point you had zero credibility. Italy had already watched you stab and unstab everyone you could; whereas France has worked with Italy all game. It's at this point in the game that it helps to have a small country in your pocket, who trusts you. It is these small countries that become critical to get the solo; unfortunately, by the time you need them, it's too late. You need to build relationships early.
1906:
Another good year for you. I'm impressed you were able to take KIE (in case it isn't clear, I'm commenting as I go). You also lucked out with Italy mistrusting France enough to fall back to ION. It also appears you're starting to build trust with Austria (by his fleet build), which I'm impressed by. If Italy and Austria got along a bit better, you'd be in a lot of trouble.
Still though, despite your progress, you aren't up much in centers and the board is starting to solidify. While annoying, those little countries are key to causing chaos. You just need them to cause more chaos to your enemies than you.
1907:
You continued to work very well with Austria. If you had a hand in getting him and Italy to fight, then well done. Losing EDI and KIE was disastrous, though. This is where having helped England early against France would have paid off: you could have cashed in on the favor. You say you guessed wrong with KIE, but I honestly don't see what your moves were meant to defend against. England taking DEN? I would have left DEN open, to be honest. There was a much better chance of England taking EDI.
1908:
1908 is where I think things really end for your hopes of a solo. You've lost England and most of Germany. France has no become a real solo threat in his own right, so you need to keep Austria as a balancing force. Italy will probably be eliminated and I don't see any way you can race France to 18. But, let's see.
1909-1915
Yeah, as I expected, you spent the rest of the time working with Austria to prevent France from soloing. A good call, by the way. With how close you've been all game, it must have been hard to accept the draw, but not doing so would have been disastrous.
Overall, I think you played an amazing game. The biggest advice I'd give is the importance of building relationships (at least one) throughout the game so you have some credibility when you need it. This can be exceedingly hard to do and I will freely admit it's a serious weakest of mine as well.
I hope this has helped. I eagerly await criticism on my analysis.