When i tried to install the latest version of forge (ver 10.13.4.1614) for minecraft 1.17.10, when  on the menu to choose the installation of client, it said "minecraft directory not found" although the bar showed the path for .minecraft and the launcher_profiles.json file was in the .minecraft folder. i am using the new minecraft launcher with a microsoft profile.

Have you played Minecraft 1.7.10 without Forge? Because if not that is probably why cause I am pretty sure you have to run the version you want to download at least once if you haven't try playing 1.7.10 quitting then installing. Also the latest version of any Forge download can be buggy as shown with a bug at the download screen. This can possibly be fixed by downloading the recommended version of Forge for 1.7.10 I am also assuming you meant 1.7.10 since the Forge version matches up.

Edited April 9, 20214 yr by FireSlash_YT
Forgot to mention a few things.

1.7.10 isn't supported on the forums. please update to a modern version of minecraft to get support