Is It A Good Idea To Use Gmail In System Emails

Hi,

Note that there are different email address options in that section of the configuration.
For the bounce email, from email settings, you want to keep an email address which is linked to your hosting / domain. Because an email server cannot send an email with a "from" email address of another domain name (you can't send emails with a "from" address of the white house and impersonate the USA president for example).
However, for the payment notification email address and the order admin email address settings, yes, using a gmail address there will help circumvent the email forwarding issue you have on your hosting (supposing that it is an email forwarding issue)