As mentioned in the other guides there is a method to make this achievement easy using the final checkpoint gltich, but if you're lucky enough to have a Series X/S you'll notice that the game loads too quickly thus makes the glitch on this small mission impossible.
The way I got around this is by moving the game install to an external HDD and thanks to it's slower loading speed I was able to unlock the achievement.
The game's total install is 74.5GB, so ensure you have a HDD big enough.
To move the game to an external HDD: Dashboard > My Games & Apps > Manage > Storage Devices > Internal Storage > Move or Copy > Select Call of Duty Modern Warfare Remastered then select Move selected > Move. This should take about 10-15 minutes depending on the speed of your HDD.
See below for the step by step method on how I did the checkpoint glitch on Series X:
1. Start Mile High Club on Recruit difficulty.
2. Get to the end where you have to save the hostage.
3. Shoot the hostage with about 2 seconds so that when it reloads the checkpoint you are back at the hostage part rather than the start of the mission.
4. Press
then Save and Quit.
5. Press the
and go down to the Call of Duty press
and select Quit.
6. Load Call of duty back up.
7. When at the campagin menu go to mission select and then Mile High Club.
8. Select Yes on overwrite save that appears and then when it asks what difficult take a breath.
9. The second you press
on Veteran quickly press
, go down to Call of Duty, press
and then quit (The same way you did at point 5 above but as fast you can).
10. Load back up Call of duty
11. Select Resume game and it should bring you back to hostage section on Veteran difficulty (Press
to confirm the difficulty at the top right).
12. Finish the mission to unlock the achievement.