Yes, specific dates are important, as shown a few verses earlier where Daniel was told by the heavenly messenger, Gabriel, that "From the issuing of the decree to restore and rebuild Jerusalem until the Anointed One, the ruler, comes, there will be seven 'sevens' and sixty-two 'sevens'..." (Daniel 9:25). This is not just an unrelated point by way of an introduction to my answer - this is germane to your question.
A few years later, but only a few verses further on in the written record, Daniel is given a vision about a future great war. By the time of the vision, it was the third year of Cyrus after his conquest of Babylonia. However, in Cyrus's first year as king, he issued the decree to restore and rebuild Jerusalem - 2 Chronicles 36:22 & Ezra 1:1-4. Cyrus's first year began on 24 March, 538 B.C.E. (This is according to a Nisan to Nisan Jewish calendar.) The significance of such details is that it can be worked out (with hindsight) that 483 years passed from the issuing of that decree until Jesus rode into Jerusalem, being acclaimed King.
Now it may become clearer as to why Daniel was inspired by the Holy Spirit to record specific details about dates. This proves that although his visions were spiritual and only seen by him, he received them in real time, during specific periods in history when historic rulers were in place.
Now, chapter 10 vs. 4 and the date Daniel gives there. Perhaps he gives it because (as he writes in verse 7) none of the other men with him by the banks of the Tigris [Hiddekel] river saw what he saw. Interestingly, akin to Saul of Tarsus on the road to Damascus, Saul's entourage were similarly stricken with terror yet did not see what Saul saw. So were the men with Daniel stricken that particular day. They were so terrified, they ran and hid themselves. Perhaps Daniel could call upon them to testify that something supernatural happened that day, even though only Daniel saw the vision of an awesome heavenly one, who spoke to him about his prayer being answered.
Daniel's mention of the fast was peculiar to him, and it followed his realisation that the foretold 70 years of exile were now up. So, he fasted for three weeks.
You correctly say, "If details are included at, or near the start of a section, initial details will be foundational to portions downstream." but my suggestion is that the details about dates in Daniel 9:25 forms the start of a section flowing 'downstream' to Daniel's vision in 10:4, and the importance of the dates connect with the 70 year's exile having ended, and a future coming of Messiah as King. Daniel's three-week fast prepares him, personally, to receive an angelic revelation, though he didn't know that while he fasted.