the likely reason why fasting later became associated with the run-up to Easter is that people started holding baptisms at Easter. The three-week long preparation for becoming a Christian through baptism included fasting, and as baptism became more strongly associated with Easter in the fourth century AD, it is possible that fasting in the lead-up became more generalised to include people who were already Christians
Makes a connection between fasting and Easter baptism
Peter I of Alexandria in the fourth century who connected Christian penitential (still not Lenten) fasting to Jesus’s 40-day fast in the wilderness:
This shows a late dating for the "idea" of a Lenten 40-day fast. It also shows how disconnected the practice is from the narrative it attempts to draw form. Initially, fasting potentially also connected with leading up to Easter baptism is now drawing from Jesus 40-day POST baptismal fast prior to temptation.
earliest reference to a sustained fast of more than two or three days is in the Didascalia, a Syrian Christian document probably from the the third century AD. Therefore you shall fast in the days of the Pascha from the tenth, which is the second day of the week; and you shall sustain yourselves with bread and salt and water only, at the ninth hour, until the fifth day of the week. But on the Friday and on the Sabbath fast wholly, and taste nothing … For thus did we also fast, when our Lord suffered, for a testimony of the three days … This text connects a six-day fast with Easter and with Jesus’s suffering, but surprisingly still not with Jesus’s 40-day temptation
Evidence of fasting of yet another different length (6 days total) and form (several days bread, salt and water). Advocates fasting on Sabbath which would be a break from Jewish tradition.
Noted the disconnect with connection to 40 day temptation and instead after a few days of bread and water re-connects to the Fri-Sat 40 Hour type fast around Jesus in the grave.
it was only after Christians began to fast specifically prior to Easter, about 300 years after Jesus’s death, that anyone looked to the Bible to find a source for the practice
Seems a bit sweeping and generalised for fasting, although could be meant to apply to the Lenten fast which is definitely a later development