In his introduction to Pirkei Avot, Rambam famously explains that all miracles are actually just pre-programmed into nature, so basically, they are all natural (or, more deeply, all of nature is a miracle). In Mishneh Torah, Yesodai Hatora 8:1, the Rambam strongly explains how a miracle is not meant to be there to enhance our belief in God.
So, in order to explain my answer, I have to disagree with your definition of "miracle" in the question.
The Torah idea of a miracle isn't the same as we understand the english word "miracle". The word is "Nes", which means "banner" or "sign". The connotation is that Hashem is raising a flag. When something happens, in nature, that points a finger to something Hashem wants us to see, that is a sign, a banner of His involvement, a נס.
As I heard from my Rav in shul this shabbat, the Maharal (חידושי אגדות - שבת כ״א:ד״ה כשנכנסו יוונים להיכל) talks about two types of נס. A Nes Nigle - an open miracle, and a Nes Nistar - a hidden miracle. This helps us understand something we say 3 or more times a day in davening: וְעַל נִסֶּֽיךָ שֶׁבְּכָל־יוֹם עִמָּֽנוּ. As the Rambam said, nature and miracles are one and the same, sometimes Hashem's signs are obvious, and they break the patterns of nature (perhaps even the law of nature, according to some opinions), and some, the vast majority, are hidden, and take the eye of emuna to see and appreciate.
If we continue this line of thought, Purim represented the beginning of the Jewish nation's full maturity, to escape the "miracle" system as part of our relationship with Hashem, and get a chance to engage in a relationship with Him without miracles, which is the ideal.
The avodah in this system is to see Hashem without the crux of miracles. The Purim story is told in this fashion. There are no "typical miracles" (clear breaks from the normal rules of nature), but that doesn't matter, you can still see that normal, routine nature is full of signs of Hashem. The story of Purim, read with the Oral Torah's commentaries, demonstrates this in many ways, which many of the answers here have already presented (and a great link collecting them is provided in the tl;dr). There are lots of things that "happen", that are seemingly happenstance, but when you are a person of emuna, and know that Hashem loves and protects His people, you put all these happenings together and are meant to come to the conclusion that Hashem was there, the Elokim, Master of Nature, watching over His people.
tl;dr: a miracle means a "sign" in lashon hakodesh. The Purim story is full of signs of Hashem and His will. They just happen to be non-miraculous (english word).
We aren't meant to view these miracles as signs that Hashem exists, according to Rambam. However, I would posit that we are supposed to view these miracles as signs that He still loves us and wants us, and our convenant with Him are eternal. Clearly, this is the "sign" that Hashem is giving at this critical part in our history.
All this was preparation for the future. The Chanuka oil marked the last obvious miracle, the last time Hashem held our hand before our big journey and mission into Galut, and ever since then, we have been plunged into a world of ceaselessly normal nature. As Jews, according to the Rambam, our job is to still see Hashem everywhere, in everything. Our job is to realise the true signs of Him, His love, and His greatness, which don't need no miracles to prove!