Why is it 'tsuitachi' if the pronunciation can only be ichi, hito, or hitotsu?
There are a few words in Japanese where the Kanji reading does not match up with the given 音読｛おんよ｝み or 訓読｛くんよ｝み readings. These are 熟字訓｛じゅくじくん｝ particularly if the reading is more important and derived from the meaning of the word and not from a combination of the Kanji that make up the word. Oftentimes, these are used for what are known as 大和言葉｛やまとことば｝, or "words native to the Japanese language", which are approximated in Kanji.
In present day, dates in Japanese are given in the pattern Year-Month-Date, as a count. For example, today is 二千十七年｛にせんじゅうななねん｝九月｛くがつ｝五日｛いつか｝ (more commonly written with numbers such as 2017年9月5日).
However, in the past, the start of the month was categorized by the start of the lunar cycle, and so had a special name. This was known as 月立ち｛つきたち｝. Therefore, 1日 (or, in this case 一日) is derived from the older Japanese word for referring to the start of the month, also known as the moon rising, or 月立ち｛つきたち｝.
月立ち｛つきたち｝ over time morphed into the word ついたち, which is written with the 熟字訓｛じゅくじくん｝ of 一日｛ついたち｝ for the sake of preserving the pattern of counting the start of the month as the first day, but also because it carries the similar meaning of the "rising of the new moon" or "the start of the month", which also is the "first day of the month".