If the menu opens but prints %player_name% literally, test the variable in PlaceholderAPI before editing the menu. Run the first command in-game on the same Paper server that hosts DeluxeMenus.
Test the variable outside the menu
If the first command returns your name, skip the next three lines. If it returns the placeholder itself, download the Player expansion, reload PlaceholderAPI and repeat the test. From the console, replace me with an online player's name and omit the leading slash.
/papi parse me %player_name%
/papi ecloud download Player
/papi reload
/papi parse me %player_name%
The test works but the menu still shows the placeholder
Check that you edited the menu that actually opens. Try the variable in an item's display_name or lore, run /dm reload, then close and reopen the inventory. If only a balance or rank placeholder fails, identify the expansion and plugin that supply that data. The Player expansion only provides its own placeholders.
If /papi is unavailable, inspect the startup log to see whether PlaceholderAPI enabled successfully. DeluxeMenus declares it as a required dependency.