Assuming you are a native Hebrew speaker (like me), I can see why you'd say that, but it's debatable. If you you want to speak "proper Hebrew", or want to write it in a newspaper or a book, then בא vs. מגיע should be pretty much like "come" vs. "arrive" in English, and here it should be בא. I believe that several decades ago all Hebrew speakers would speak this way, too.
In recent years מגיע has been gaining ground over בא by Hebrew speakers (for "come"; בא has been gaining its own ground over other words in different meanings...). So now indeed you're more likely hear האוכל מגיע. But I hope it's not yet a sweeping victory. FWIW I would still say האוכל בא - at least I hope I would, I could be misleading myself...
Great but immagine teaching a foreign person the word "action" as "akshen" just to show him how the word is pronounced, i mean you don't have the right to change the spelling of a word just to let a foreign read well, those thing must be learnt either with "niqqud" or with "audio". For me the only correct spelling is אכל.
It's not the same as in English, and has nothing to do with helping foreigners. You may be missing the fact that modern Hebrew is written mostly without niqqud, not just as colloquialism, but in the most respectable press, literature, official documentation, and, to nail it down, even in the website of the Academy of Hebrew Language. Niqqud-less Hebrew has well-established rules and habits. Adding the vav is not just to help foreign readers, it's to help native Hebrew readers. Writing אכל for "food" is a plain error.
There is no technical problem with saying "אוכל בא". The problem is contextual and it might not be understood, and sounds extremely awkward on its own. Here is why: Hebrew has no 'progressive' tense (like most languages), and it is derived from the context.
So if you translate to English, "האוכל בא" would like "the food is coming" (although also awkward, as I pointed out in my previous comment), while "אוכל בא" would be more like "food comes". You can see that even in English, saying this separately without any context would be extremely strange.
The "ה" before a noun is used as the word "the" in English. You'll find later that prepositions are also witten with single letters, for instance "ל" before a noun is the word "to". "Is food coming" would be said "אוכל בא?" When being specific about which food though, you would say "Is THE food coming"? Which is האוכל בא?