To resolve that issue:
1. Open the Minecraft launcher.
2. Click on "Parameters" on the low left side.
3. Click on the "Account" tab.
4. Click on "Change account", the XBox window to change account will open.
5. Click on "Change account" in the XBox window.
6. Select your current account (mail address). As the account is basically the same (only the gamertag has changed), XBox will update the link between the Minecraft launcher and your XBox/Microsoft account (with the new Gamertag).
MCL-22870
To resolve that issue:
1. Open the Minecraft launcher.
2. Click on "Parameters" on the low left side.
3. Click on the "Account" tab.
4. Click on "Change account", the XBox window to change account will open.
5. Click on "Change account" in the XBox window.
6. Select your current account (mail address). As the account is basically the same (only the gamertag has changed), XBox will update the link between the Minecraft launcher and your XBox/Microsoft account (with the new Gamertag).