Hello ShineMoe,
Do you have any account that is currently sign in when you tried to delete old gmail account?
To remove the gmail account you want, you have to sign out first all the the account signed-in on your chrome. once accounts are signed out you can then remove the account that you want to remove. click on your gmail account icon then click sign out all accounts. click on the drop down arrow beside the gmail email address then click REMOVE. then the gmail account will be removed.
More details on this link:
https://support.google.com/mail/thread/4524465?....
You're response is highly appreciated. Please let me know if you need further assistance.
Sincerely,
Antonio