Your concern would be what rotted out the threshold in the first place. To do it right would entail pulling the whole shebang in order to properly flash the area of the threshold and address the cause. You could possibly cut the threshold out while tha jambs remain in place with a flush cut saw, but you are left with having to replace it with the exact sized and profiled threshold which might not be an easy find.

I'm guessing you do not have a continuous threshold, but that there's a break where each of the sidelights meets the door's threshold? We had a similar situation with rotting because moisture wicked up in the gap or break area. Now they make continuous thresholds to avoid this problem. Anyway, we ended up replacing the whole unit- doors & sidelights. Costly and a hassle. We did have some minor rotting under the threshold that required minor repair.

It is usually not very hard to remove the entire frame.

Remove the interior trim, cut the nails or drive then through the frame completely, then just pull the whole thing out.

This will allow a complete inspection of the threshold and the joists under the door.

Install whatever flashing is likely missing, make any other needed repairs (including the unit itself) then put it back in the opening, square it up and nail.

If the job goes over more than one day plywood can be placed over the opening.

I agree that you should look at what caused the rotting - we had a similar situation and intended to replace the door/frame/threshold, or so we thought. The end result was that we had to replace a major basement beam that had extreme water damage. Thank goodness we addressed it or that side of the house may have shifted/fallen! We had the contractor build an awning over the door to push the water away.