A Result of Misguided Calculations
The chet haegel was not a straightforward case of idol-worship. It was shortly after matan Torah, and Klal Yisroel were in fact on a very elevated spiritual level. The cause that led to this terrible sin, was that they made too many calculations and came to their own conclusions. They did not follow the guidance of Aharon Hakohein and the ziknei hador.
The Ohr Hachaim expounds greatly on exactly what the yidden were thinking, and how it led them down a path which ended with the terrible sin of the golden calf. There were very few who actually explained the calf as a G-d, however there were many other classes of Yidden who participated in the sin on different levels, and had different calculations and reasoning as to why they created the golden calf.
The Yidden saw that Moshe, who was Hashem's emissary to them, was missing. They thought they need to create a different vessel to bring הַשְׁרָאַת הַשְּׁכִינָה into their midst. It was a tragic mistake which resulted from too much independent calculations, reasoning and misguided reckoning.
The Beis Halevi is one of the chief commentators who explains the connection between the Parah Adumah and the chet haegel in this way. He explains that all the mitzvos have taamim, both straightforward understandable explanations, and deeper, loftier reasons which only very holy people understand. The seforim tell us the extensive effects of every single mitzvah we perform, and how it influences phenomena in the upper-worlds. For instance, the מְלֶאכֶת הַמִּשְׁכָּן brought about הַשְׁרָאַת הַשְּׁכִינָה, and even the minutest detail that was part of the process of preparing, building and serving in the Mishkan, was instrumental in increasing the הַשְׁרָאַת הַשְּׁכִינָה. Every specific aspect of every mitzvah has a role in the impact of the mitzvah.
The Beis Halevi says that for instance, a chochem may mistakenly think that he understands the reasoning behind every element of the mitzvah, and he can achieve the desired result through alternate means; he thinks he can craft his own course to holiness. If he knows the סוֹדוֹת and תִּקּוּנִים, the deep secrets and restorative powers of the mitzvos, then he may chas vesholom devise his own strategies on how to achieve them.
That was the fatal error of the chet haegel. When some of the Yidden saw that Moshe - who was their spiritual guide - was missing, they rushed into creating for themselves a new way to achieve holiness. They wanted another channel through which to form a connection with Hashem. It was this misguided mindset which caused them to act upon their own understanding of events.