Yes and no.

Mars atmosphere is much thinner than earth's. Gravity is already the hardest part of getting to orbit from the ground on both planets, on Mars the atmosphere component is smaller.

The bulk of acceleration to reach orbit isn't the up part, it's the sideways part. By having a low-hanging and sub-orbital "docking point" at the base of the elevator, you are still conserving a lot of fuel and delta-V even though it doesn't go "the whole way".

Also, having an asteroid anchor point gives space elevator architects more freedom. Unwinding the inner and outer tether doesn't need to be perfectly synchronized, and tether lengths and counterweights could further be used to adjust Phobos' orbit around Mars.

None of what I just said is fast or easy, but when we're talking about space elevators nothing really is. As described, this might be the "beta version" space elevator that's deployed before any are used on earth - lesser risk, more room to experiment and learn, still has some use if we're jacking around on Mars, etc.