What you pick for your personal email is entirely up to you since the only people that will see it are those that have moved into your out-of-work life.
I think this is a non-issue, for reasons that no one has answered yet.
When you have your email forwarded to your Gmail, you can also set up Gmail as the sender for your other email address. This means that you can read mail to your EDU address in Gmail, and also send emails and reply to emails so that the from field is your EDU address.
The configuration of all of this is beyond the scope of academia.SE, but there are plenty of other sites in the stack-exchange network where the question has probably already been answered.
In terms of the non-academia question about professionally of Gmail in the workplace: it is completely fine to use your Gmail for personal communications. Anything professional should go through the domain name of the company or institution that is employing / sponsoring you.
If your main work is at one university, but you're doing residency at another and have an email address there, most of the time you probably will still use your main institution's email. The exception to this rule guideline is when you need to communicate with the administration about needs that are under their 'jurisdiction' (facilities, scheduling, etc).
The fact that other colleagues use a non-professional email account for their communications does not mean it's good etiquette. It could (and likely) just means that they are too lazy or too lacking of knowledge to properly setup their email systems for proper separation of professional and personally messages (while also remaining convenient). You can do better than this. From this, what you pick for your personal email is entirely up to you since the only people that will see it are those that have moved into your out-of-work life.