Is a standalone periodic email checker app feasible?

I bought my first smartphone in 2012. I had one motivation for doing so: my clients, instead of ringing me, were increasingly emailing me. I had to have a computer "to go" to respond in a timely manner. But a flaw emerged as I got to grips with my first Samsung: although I had configured a bleep to be sounded every time an email arrived in my Gmail inbox, this, er, didn't always happen. At that time, I think, I probably assumed that this was a small thing which was associated with the teething troubles of a spanking new technology, and would quickly be ironed out. That did not happen. 12 years later and no improvement. It is simply unbelievable. I have spent many, many hours viewing forums about the notification problem and struggling to try anything and everything, repeatedly. Often finding that supposedly available options etc. didn't in fact exist on my phone. The defect is still there. I almost said "the Samsung notification problem". Indeed I almost also said "the Android notification problem". Recently I have even contemplated the unthinkable, i.e. going over to the **Other Side**. But when I googled things, I found that (reportedly) various models of iPhone in reality seem to suffer the same thing. A few days ago it occurred to me: is there an app for Android which has one single function? Every 5 minutes, it looks at the Gmail inbox for a given account, and if an unread message is present it makes a bleep. How hard can that be? I searched for such an app at PlayStore. I didn't find one. Which doesn't mean one doesn't exist. If someone knows of such a thing, please enlighten me. Therefore my question is: **is it feasible to create such an app?** I have never tried to create an Android app, but there's always a first time. Does anyone know whether emails arriving in a Gmail inbox are detectable by the phone pretty much immediately? My understanding to date has been that this giant intermittent defect happens despite the fact that a new email has already arrived in the phone. Maybe that's not always the case. But certainly there have been MANY MANY times over the past 12 years when I have glimpsed down at my phone to find, yes, an email has arrived in the Gmail app ... but my ****ing phone never bleeped me a notification. The app I imagine would at least put an end to that source of fury. This is no joke for me. It hasn't been for years. With my line of work, my clients will tend to contact other people if they don't get a response within 30 minutes or an hour. I have lost 000s of £/€/$ over the past 12 years due to this unfathomably frustrating nonsense.