Is It Okay to Continuously Use One's Private Email Address?
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.