How can I destroy the flag without falling to my death?

I've never had much problem. I have various methods of climbing ships.

Usually if I have to capture a flag my tactics are after boarding starts

1. Use the swivel gun to do what damage I can.
2. Use the quick elevator thing on my own ship to get up to the platform.
3. Shoot enemy soldiers on the central platform (or others).
4. Swing across to their boat on a rope and assassinate people and then fight generally til I am at or close to the soldier death limit (if one or two short I hope that my crew will finish the job).
5. Use the quick elevator on their ship to start my ascent. There is of course no soldiers up there any more.

This has never really failed me. I do occasionally skip some steps but the above is, in my view, pretty efficient and pretty foolproof.

For Frigates and Man-of-Wars, I can usually get onto his platform and fight him in melee

1. Use an elevator on your ship
2. Wait for the grappling hooks to pull the other ship completely next to you
3. Freerun across the rigging to his ship and disarm him.
   On a Man-of-War the sniper may attempt to fire at you (indicated with the crosshair in a red circle) If this happens run all the way to the other side of the ship underneath him before climbing up on the rope.
   
   
4. Use his musket to shoot the other sniper

From there I find the quickest way to get to the top is to go back onto the wooden beam and use the ropes stretched from one level to the other to run up to the top.

I never have problem. I climb. If the guy with snipper attack, I just counter. I think this is the most unrealistic aspect of a game. How can a guy from the bottom attack up against someone ready with gun. Yet again, I never have problem.

Now there are usually 2 snipper. That's not a big problem either. Kill one melle. Face the other snipper till it's selected. Select gun. Press triangle to shoot. One shot, with the most expensive gun and the guy is death.

This happened to me the first time I was trying to kill the scouts.
Now, I always know those scouts are there, whether I have to kill them or destroy the flag.
Knowing this, I use the lift and counter the scout's attack the second I get up on his platform.
I typically then throw him off the edge like he would have done to me, because it amuses me.

You need to ge to the highest mast, then simply jump straight up to grab the last platform. Then destroy the flag. So simple,but deceiving.

To answer the actual question, there is no way to climb the mast without a chance of falling. This is because the freerun system falls apart for some reason and Kenway will jump to his death even without pressing (x) (or whatever the jump button is for your system) while holding the freerun button. Normally holding the freerun button guarantees that Kenway will not jump a distance that will cause damage but again, for some reason, he will jump from the top of the mast to the deck and kill himself, even without performing a "leap of faith" jump.

-yes I am playing a game from many years ago and commenting on its flaws far after any opinion of the game matters. Thank you.