So, I've now played through two more games with the "no beelines" house rule. Here is an update on how it worked out.
Rome
Rome is a little more -- irritating -- to play, since you have to wait significantly longer before you can get to iron working to see how many legions you can build. I had a couple of false starts where the entire continent ended up with only one 2-iron spot. But on my third try I was able to claim a 6-iron spot (in a location next to Egypt, which was guaranteed to provoke them to DoW) with the same number of warriors ready to upgrade. By the time I hit the industrial era, I was clearly the runaway civ. Going wide with Liberty for Rome works very powerfully with no beelines; you actually get the time to use all of the early-era units for a while.
Ottoman Empire
In response to @smallfish's comment, I then tried a game as Suleiman. This was harder, but part of that may have been attributable to the starting position, which was up near the tundra. I did manage to get the "aurora" pantheon, but was limited in my ability to expand early because Napoleon was my nearest neighbor. I had to fight a tough early defensive war, which turned into an opportunity to capture one French city when Napoleon decided that fighting two wars at once (by DoWing Spain) was better than concentrating on winning just one. I then had to bide my time until I got jannissaries before I could attack. By that point, Nappy had DoWed everyone on my continent (and eliminated Spain), so Germany and Siam remained friends with me while I reduced France to a civ that only controlled one tiny snow-bound city.
Meanwhile, on the other continent, the Iroquois and Mongolia had eliminated the Songhai, and didn't much like each other. I tried being friends with them both for a while, but that clearly wasn't going to work in the long run. Since Hiawatha was clearly a threat to sneak off to a science victory, I eventually sided with Mongolia, and got them to sign a defensive pact. Hiawatha went ahead and attacked Mongolia anyway, which brought me into the war. Interestingly, both Mongolia and Germany remained in DoF's (and RA's) with me for the rest of the game. (Siam was small and irrelevant, but decided to join the Iroquois-France faction.) My war against the Iroquois was fought entirely with naval and air units. (The goal was simply to weaken them enough that they would be unable to get a science victory.) The Ottoman UA may be stronger at the point where destroyers start becoming available, especially when the enemy still has caravels and ironclads around. When you capture one of those ships, you send them home to promote them and regain health, and they come back with the prize ship ability themselves. Eventually, I had the choice of a diplomatic or science victory, and took the diplomatic one just because I felt like it.
Overview
My overall impression at this point is that the"no beelines" rule leads to more interesting games. However, in every one of them, I've had a tech lead by the industrial or modern era that just became more and more overwhelming. So there hasn't been an opportunity to fully use some of the modern units. Although the last war against the Iroquois had a little of that, I never used a tank, helicopter gunship, rocket artillery, or stealth bomber....
If you're not looking for the fastest victory, the "no beelines" rule may actually be fairly strong---at least at king or below.