This site states:
The minhag of many Chassidish kehilos is not to say tachanun at mincha. This minhag is recorded in many Chassidic and Sefardic sources. A number of explanations are offered:
1 they often daven until after shkiya, and according to many opinions tachanun may not be said after shkiya, so a blanket rule was instituted so as never to come to saying after shkiya, which in some kabbalistic sources danger is associated with this practice.
2 tachanun by mincha requires intense concentration, which most people don’t have in the middle of their day
3 after it is night in Israel tachanun should not be said even in other parts of the world
In numerous siddurim that I have seen, there is an instruction next to Tzidkatcha Tzedek (said after the Chazan's Amidah repetition of Shabbat mincha) that says that Tzidkatcha Tzedek is omitted on occasions when tachanun would be omitted. I understand this means Erev Yom Tov, Hol Hamo'ed, Rosh Hodesh, etc. But, would this include those places that never recite Tachanun at mincha, as well?
In viewing the above reasons, I can see not saying it if either it is after Sheki'ah in your local area on Shabbat, or if you wish to follow rule #3. (According to rule #3, it seems that in North America, you'd never be able to say it as even at the earliest time allowable for mincha, it's already night in Israel.) But rule #2 seems vague. I don't know what the "intense concentration" aspect is, here. Do they mean specifically business concerns or is it something else?
If places do / should say it, please explain why this is not a "contradiction".