[Native Rus Speaker] I can't understand why you write borsch instead borsh. Sch means Щ only in Deutsch, doesn't it? Maybe it's because of relationship of Rus and Deu, because of it all Russians say Г instead Х in the beginnings of the words: Houdini - Гудини, Hitler - Гитлер, Harlem - Гарлем etc.
That is actually wrong. English "sh" sound is much closer to Russian "Щ" than to Russian "Ш". Meantime English does not really have an analogue of Russian "Ш"; you can read a long discussion of this issue in this thread.
In fact, the only significant difference between English sh and Russian щ is that Russian щ is a long consonant (while e.g. Russian ш is not). So my guess is that the English transliteration of "борщ" piles up all those consonants to indicate the long nature of щ. Otherwise "borsh" would do just fine, and there is definitely no "t" sound in that word in Russian (I've seen it spelt as "borscht").
G for H does seem a bit odd, given that Russian has Х - and the translators used it for Sherlock Hhholmes and Mrs Hhhudson in https://en.wikipedia.org/wiki/The_Adventures_of_Sherlock_Holmes_and_Dr._Watson. But there are quite a few of these to get used to - my Russian wife confirms that Э really does get Russian speakers closer to the English pronunciation of Andrew than А would.
Some people? Борщ without (red) beetroot is щи, not борщ. Some people might still call it борщ, but I would bet the same people would refer to the sludge brewed from acorns as coffee (according to my grandmother, this is what they did in the lean years after the second world war). I can see how the original question ("Это кофе или борщ?") might actually sound meaningful to those people ;-)
While apparently the Wikipedea agrees with you: https://tinyurl.com/ycrq7agl, it would never occur to me to call the sorrel-based soup as "зелёный борщ" -- only as "зелёныe щи". And it's actually one of my favourite soups, so this would definitely not be due to the lack of familiarity. So I wonder if this is regional: for me, the all-encompassing word is "щи" while "борщ" is strictly beetroot-based.