Outlook web is still better than gmail for work-related-use, in my opinion, let alone full-featured outlook.
I work for one of the larger pharma companies... they made the migration to gmail for work last year and it's abysmal... everyone hates it.
I genuinely prefer it for personal use where i'm not so desperate to keep track of things.
But for work related use... it's just... terrible... Microsoft are better at this one.
(and I don't like microsoft very much

)
For OP... there is nothing wrong with a gmail address... it's free and common, it's like having an AOL address a decade ago... nothing special and nothing taboo.
I still use my
[email protected] address for job applications/CV
I have been thinking about moving to using my own domain, I already own firstnamelastname.com, but am not that fussed...