The use of a sextant by itself isn't the limitation, it's having to do calculations by hand instead of by a computer.
I agree with uhoh's comment that the final approach can be done manually, without either the computer or the sextant. For example, the extraction of the Apollo lunar module from the 3rd stage could easily be done manually. I saw an article that stated that manual docking can be done within a distance of 0.1 km, but I can't find that article again.
The real problem is getting close enough in the first place. Consider the launch of a lunar module from the moon, trying to rendezvous with a command/service module. The CSM has an orbit with a particular altitude, eccentricity, and inclination. The lunar module needs to get into a matching trajectory, all while under the influence of the moon's gravity. This article explains the details of this process. Yes, you do need the position and velocities of both vehicles, and a sextant may help you get those. But there's far more to the calculation than those parameters. Most importantly, note the following near the bottom of that article:
Calculating the velocity needed for this maneuver is more than simply raising the pericynthion, as other orbital parameters, plus maintaining the delicate relationship between the CSM and LM need to be taken into account. These complexities are beyond the capacity of the limited memory of the LM guidance computer. Rather than selecting a specific program to perform the Boost maneuver, the calculations will be performed in the Real Time Computing Center (RTCC) and relayed to the LM crew. The crew, in turn, will use one of the "External Delta V" programs to perform the maneuver and manage this burn. Most of the other programs used in the LM (such as descent, ascent and rendezvous), have the guidance and navigation solutions calculated internally; that is, using only the resources available on the spacecraft.
If such calculation is too difficult for the on-board guidance computer, it's unlikely that the astronauts will do any better with pencil and paper. And a launch from Earth is even worse, as you also have to account for atmospheric effects.
Suppose you are lucky enough to get into the same orbit as the other vehicle, you're "behind" the other vehicle, and you want to catch up. If this were a car on a highway, you'd just speed up until you catch the other car. However, the article points out why this won't work:
If you speed up, it will push you out into a higher orbit. By Keppler's laws, this higher orbit will have a longer period than that of the other vehicle. You will therefore actually fall further behind the other vehicle!
Perhaps an astronaut could practice in a simulator, but it's hard to make a realistic simulator without a computer.