Swordsmen come into the game when you get IW and that is the next tech after BW if you beeline to it.
I see your point going with an early axe-rush but this can only be done in an ideal situation (copper very close to capital).
In this situation (very early) most enemies will likely have no ressources online and the only-axes strategy is OK.
The more common situation (2 out of 3 games in my experience) is: copper not very close or even no copper visible, then you have to go for IW straight away.
Yes, IW is right after BW, but you sacrifice teching in a more beneficial direction (cough alphabet cough) in order to get it, plus the enemy capital is probably getting buff on culture and maybe a wall by the time you march there with enough metal-wielders to take his/her capital. I've won doing that, though, just rushing no matter what the tech cost, getting that precious copper or iron ASAP. But lately I've found an alternative solution:
Since I never bother with archery, always going for axes for both anti-barb and city-rushing, if I don't have copper decently close but have stone decently close (which happens fairly often), I immediately turn my attention to building Stonehenge + Great Wall + Pyramids in my capital, helped by some chopping, and REX to 4 cities while sealing off at least one flank by placing my cities appropriately. If you do it quickly enough, GW pops up before barb axemen show up, and Pyramids doesn't take that long, maybe 15-30 turns on Epic if you have stone, the variability depending on how many hills are in your capital's fat cross. It goes by even faster if you chop. Representation gives you unparalleled tech points because your cities can keep growing well beyond normal happiness limits and thus work lots of cottaged tiles, and then the parade of specialists from your 3 wonders just keeps adding to the fun. While cottage spamming behind the safety of my GW and not spending anything on military (I pay a LOT of attention to diplomacy and religion and always give tribute if it is even barely reasonable--and I can always whip out some chariots if someone sends axes my way), beelining to alphabet, math, currency, and CoL, and trading a lot early on (I'm usually the tech leader early on with this strat, because nobody else has alphabet and I keep it that way as long as possible so all trades have to go through me), and I usually found Taoism for a shrine later on. Pyramids + GW also tends to give you a GE which you can spend on something nice. A GP is decent just to simply settle for gold and beakers (from Representation) if you don't want to or can't shrine.
In my current game I got an early GE which I used on the Great Library, which combined with my super-science/wonderville Bureaucratic capital and Representation = I never lost the tech lead. Hell I even got another GE soon after that, thanks to the wonders and specialists working together, which I used on a Parthenon because there wasn't anything better to build. That just spammed me some more Great Scientists. I even found time to build the Uni of Sankore just for the GS GPPs and not its wonder-ability, because I couldn't safely have a religion for quite a long time, thanks to the warring religious factions around me.
After I get currency and CoL, it's time to invade my neighbor. In my current game that invasion force was just some axes and spears since I was so freaking far away from unoccupied iron. I didn't even have ivory within easy reach, but whatever, the axes were good enough to raze some cities and pillage like crazy, sustaining my tech rate to get construction quickly, and you can guess what happened after that. I also never did get my Great Prophet until pretty late into the game, so my holy city just sat there unused, but that's for the best anyway since I'd rather have an early GS or GE than an early GP.
In other games where I did the three-stone-wonders combo, I've done other things and won them all easily. In the least-easy game, I got attacked early by my supposedly Pleased neighbor Ragnar (I even gave that lousy guy tribute), but repelled it thanks to my REXing near metal, which enabled me to whip out some axemen, after which I ran over Ragnar and the guy behind him as well (whom I worker stole from early on since he was very unpopular with everyone anyway). My other games were really easy because I wasn't behind in tech as I normally would be with an early rush. Bribing a few civs to attack each other once you get a small tech lead over ANYONE with a big army (Napolean is my favorite; he usually starts wars by himself anyway; but even if he doesn't, he's usually got a huge army and is thus a bit behind in tech, making him a great candidate for bribing to attack someone) = you won, because sooner or later those "you declared war on my friend" points add up as your original war spirals out of control, especially if anyone dogpiles. Just about any war not involving you is good, because you can assume that the combatants aren't going to start a war with you while they've already got one on your hands = just cultivate good relations with those civs who aren't in the war and you can keep building and teching with abandon. I've even gifted a ton of units to my proxy-army before, just to keep a war going, my record being 34 cavalry, as many as 50 Redcoats, and an assortment of siege weapons to Hatty, whom I was using to "fight" the map leader. She lost a city and was starting to really get beaten up, so I helped out. My power rating fell by half but hers had a huge spike, and a few turns later she recaptured her city. Coincidence?
So um, no, you don't have to go for IW if you can't find close copper early on.