I would have a question: Is it okay to use one's private email address to reply to all the university-related emails after having set up a forwarding to one's private email address - be it as a Professor, PostDoc, PhD-, graduate or undergraduate student, and use one's university/work email only for the most important emails that one wants to send?
There might be internal regulations about e-mail use at your institution; if there are none, then I would say: It is your own personal choice.
Even in academia, many people do use their non-institutional accounts because - to cite some reasons from this source - they provide more functionalities, are sometimes more reliable, enable greater storage space, support larger attachments, last longer over career trajectories, and may have a less rigorous spam filter.
There are some interesting studies in this regard:
The provision of public e-mail accounts (e.g. gmail) for correspondence in published research articles will not affect a publication's impact (in terms of citation counts). See here.
Within the research community, there is an increasing trend towards using public e-mail accounts rather than institutional ones. See here.
Someone who conducted one of the studies cited above nevertheless recommends the use of institutional e-mail addresses. See the last paragraph here.
If you do not have a permanent job, I recommend using a private address. As a researcher, using your private, branded email address makes you easier to find when you change jobs. Some IT departments do not understand that when researchers change jobs, the research continues. They will turn off your email when you leave.
Expect to get an occasional complaint. A few universities forbid use of private email addresses. All the ones I have experience with made it easy to automatically forward official emails to a private address.
In addition to the question of university regulations, I see two aspects of this practice:
- I think it is fine to use the personal email address for correspondence concerning your research. Some people use this to have a more "permanent" email address. Of course the situation is more complicated if you want to use it to send data from studies which might contain personal data - then the next point applies.
- For the communication with students, I would not use the private email address. In the European Union, you might run into problems concerning the GDPR, i.e. the data protection regulations. This applies in particular if you are using services operating outside the European Union or free services which scan your email. Similar regulations might apply in other places. So for everything containing personal data about other people than yourself, I would only use the email address provided by the university, since they should take care of the data protection requirements.