Will photosynthesis based carbon sequestration always be more efficient than other chemical means - in terms of rate?
Rate.. no.

Brute Force

We can treat this as an industrial chemistry question. The brute-force method for carbon sequesteration is simply:


Extract CO2 from the air (using the acid/base chemistry, Sodium Hydroxide will scrub it as fast as you can supply air)
Reduce CO2 to elemental carbon and oxygen. The exact details are not actually important - that's one for the industrial chemists - just the concept that it would take a LOT of energy. Basically as much energy as we currently get from fossil fuels, and then some, to extract and reduce at the same rate.
Store the carbon somewhere safe, where it isn't going to catch fire. As long as it doesn't, it's stable on geological timescales.


The rate of this process is dependent on how much energy you are prepared to throw at it. A fleet of several thousand nuclear reactors running 24/7 on this problem would do it.

Burial

Well, brute force sounds a bit expensive.. so we can try the approach mentioned by Jack R Woods. In this case, we capture CO2 (this still takes energy, although not much in theory), and pump it into thick basalt formations such as the ones in Iceland. Many flood-basalt igneous provinces have the combination of low-silica rocks and high porosity that are ideal for this kind of sequestration. The rate is limited by how fast you can capture CO2 out of the air.  You could imagine using a Solar Updraft Tower - this would give an air flow that you could extract CO2 from, and provide power for the process. If the reaction goes as planned, it's safe on geological timescales.

The problem is, as ever, getting enough resources to build these things on the scale required.