Ultimately an elevator is going to be more efficient because it doesn't have to deal with gravity losses.

Let me pose a question to you. What does it take for a rocket to hover in place like Blue Origin's New Shepard?

If you've watched any of their launches you know they don't shut off the engine completely but keep them running the whole time while hovering and even if you haven't watched one of their launches that is the intuitive answer. If you turn off the engine completely you start falling back to earth. This same thing is happening the entire time the rocket is climbing it's way up to orbit and we call that energy used up Gravity losses.

Because of gravity losses a rocket can't take days or weeks to get to orbit as you'd burn far too much fuel "hovering" the entire time. This is why rockets are big and powerful as they need to get to orbit fairly quickly. In minutes not hours or days. This means they have to use engines that can meet those characteristics. Powerful and light weight. At the rate rockets consume fuel beamed energy really isn't practical.

We have much more efficient rocket engines that produce a lot more Delta V (change in velocity) for the same amount of fuel and could work on beamed power or actually use Solar power but the amount of thrust they produce is far too low and they often have a thrust to weight ratio of less than 1 even just for the engine which means they couldn't even support themselves to hover let alone lift a large rocket. These types of engines are used by Sats in orbit for station keeping or moving around to different orbits.

A space elevator, if we could build one, opens up many options for removing the weight of the fuel from the payload or even simply take a slow but more efficient approach. On a tether you can just stop and hang there and not use any power. A simple electrical motor pulling you up the cable and solar panels or beamed power would work. You don't even need the solar or beamed power to provide enough constant power for the climb as you could include some batteries to allow you to charge for a while, climb a bit while draining batteries and then stop to recharge them. Heck even if you had an engine on the crawler with some tanks of gas to power the climber I'd still expect it to be more efficient than a rocket.

Ultimately an elevator is going to be more efficient, because it doesn't have to deal with gravity losses.

Let me pose a question to you. What does it take for a rocket to hover in place like Blue Origin's New Shepard?

If you've watched any of their launches you know they don't shut off the engine completely, but keep them running the whole time while hovering and even if you haven't watched one of their launches that is the intuitive answer. If you turn off the engine completely you start falling back to earth. This same thing is happening the entire time the rocket is climbing its way up to orbit and we call that energy used up gravity losses.

Because of gravity losses a rocket can't take days or weeks to get to orbit as you'd burn far too much fuel "hovering" the entire time. This is why rockets are big and powerful as they need to get to orbit fairly quickly. In minutes, not hours or days. This means they have to use engines that can meet those characteristics. Powerful and light weight. At the rate rockets consume fuel beamed energy really isn't practical.

We have much more efficient rocket engines that produce a lot more delta-v (change in velocity) for the same amount of fuel and could work on beamed power or actually use solar power, but the amount of thrust they produce is far too low and they often have a thrust-to-weight ratio of less than 1 even just for the engine which means they couldn't even support themselves to hover, let alone lift a large rocket. These types of engines are used by satellites in orbit for station keeping or moving around to different orbits.

A space elevator, if we could build one, opens up many options for removing the weight of the fuel from the payload or even simply take a slow, but more efficient approach. On a tether you can just stop and hang there and not use any power. A simple electrical motor pulling you up the cable and solar panels or beamed power would work. You don't even need the solar or beamed power to provide enough constant power for the climb as you could include some batteries to allow you to charge for a while, climb a bit while draining batteries and then stop to recharge them. Heck even if you had an engine on the crawler with some tanks of gas to power the climber I'd still expect it to be more efficient than a rocket.