I just talked through these four words with my wife, who grew up in the Toronto area but has English (U.K.) parents; she’s somewhat mid-Atlantic I suppose.
For her, the vowels in “butter”, “mother”, and “father” match my accent (the “tt” in “butter” is completely different). The “o” in her “bother” is a bit shorter than the “a” in her/our “father”; seems closer to “father” than to “mother” though.
I played around a bit with speech editor/synthesiser Praat that Dr. Geoff Lindsey (who is also a youtuber) wrote about here: https://www.englishspeechservices.com/blog/the-vowel-space/
When I map vowel-like sounds (by F1 and F2 formants produced by Praat) to Dutch vowels, and I compare that to the diagrams Dr. Lindsey created with it for English vowels, it seems that several English vowels, among which the /ʌ/, are sort of in a “forbidden zone” for Dutch vowels:
Perhaps this explains a bit why it blows my mind that /ʌ/ (which in my mind is close to /ɑ/) can cross that “forbidden zone” to become confused with /ə/.
I think this identity is the cot-caught merger, and these two vowels are identical in mine as well (USofiAn midwest), while the other one (mother) is essentially ʌ
Trying to figure out how to phonetically spell my pronunciations of words that are caught in the cot-caught merger is an activity I find very distressing: English vowel phonology kind of stops making sense to me as soon as you collapse /ɑ/ and /ɔ/ and /ɒ/ into a single phoneme, but that seems to be what the speakers of our dialect have done.
Edit: failed to turn the alpha ɒ on first posting.
Interesting, and plausible. Because English often spells this vowel in monosyllabic words with a “u”, I always thought of it as a turned “v” when I was first learning it. Since the IPA is supposed to be international and unbiased (heh), I accept this was just my personal interpretation.
I thought/think of it as a turned stylized u
That never occured to me, perhaps because “u” in Dutch spelling is pronounced as /ʏ/ or /y/.
But I suppose English speakers would think of an inverted “u” when seeing the symbol ʌ, because “u” in English spelling is often pronounced /ʌ/, as in “but”.
Also, IPA seems to derive partially from the 1892 version of the Romic phonetic alphabet which contained the symbol ⟨ʌ⟩, and that alphabet was proposed by Henry Sweet, who was an Englishman.
However, Sweet used ⟨ʌ⟩ for the vowel denoted in IPA as /ɯ/ (which sounds somewhere between /w/ and /u/ to me).
For the vowel in “but” he seems to have used ⟨a⟩, which he also used for the vowels in the German word “Mann” and the French word “patte”, which I suppose corresponds to /ɑ/ in IPA.
So the history of ʌ seems quite convoluted.
And of course, in any event, at least 50% of the reason ʌ was picked was because it could be made with the piece of type of a v 
I would like to communicate while enjoying the differences, rather than forcing myself to conform to native speakers 
Careful what you wish for 
A friend of mine had worked for a British phone-support company. He was faced with a bewildering array of accents from Ireland, to Scotland and Wales, Newcastle and South of England and who knows were else. A lot of the times he could barely understand them because those were people that were in some distress and already had a problem than needed solving. That tends to make someone speak faster or louder making things more difficult. Let’s just say that he didn’t last too long in that job. 
Thanks! The map is useful for me 
It reminds me of a video about the fun of English in the UK. I was really surprised to see that they cannot communicate even in the same country council 
Dutch (オランダ語) dialects. Note that the pink colors at the very bottom are French dialects, not Dutch. And the blue colors in the upper middle are Frisian dialects. Frisian is considered a separate language from Dutch, although it is heavily influenced by Dutch.
This is a fairly fine-grained map, but I think you could even go more fine-grained and distinguish dialects of different cities that are categorised as the same dialect in that map:
My son studies in the university city of Leuven and students there come from various regions of Flanders (the Dutch speaking part of Belgium, the lower 30% of that map).
It’s not uncommon that some of his fellow students have a hard time understanding each other’s regional accents/dialects, though they can usually understand my son’s accent/dialect fairly easily. My son is from the southern Netherlands and I suppose his accent/dialect is close enough to standard southern Dutch as spoken on Belgian television.
Not only does pronunciation vary between different regions, there can also be regional differences in vocabulary.
For example, the sink of a bathroom is called a “wastafel” (lit. wash table) in standard northern Dutch (as spoken on television in the Netherlands). But in the Kempen region (northern Belgium, bordering our village in the southern Netherlands), they’d call it a “lavabo” (via French from Latin for “I wash”). Then again, in Leuven (central Belgium) they apparently call it a “pompbak” (lit. pump bowl/reservoir).
When you’re unfamiliar with such vocabulary differences, it can be difficult to understand what they’re talking about, especially when they have a thick accent on top of it and speak quickly.
I’m assuming you’re Japanese because your name “koyama” looks like the Japanese surname “small mountain”, and the video you linked is in Japanese.
Japan is much bigger than the Netherlands and Flanders combined, and I think the Japanese language has a long history all across Japan, so I’d expect there to be even more variety of regional accents/dialects than we have here.
Can someone from a rural place in Kyūshū communicate easily with someone from a rural place in Hokkaidō? I think the youtuber in the video you linked says something about that near the end of the video, but my Japanese is far too limited to understand what he’s saying exactly.
