To put this simply, setting a prior distribution for a logistic regression using a mathematical model based on background information is no more „subjective“ than deciding to run a logistic regression in the first place. That means no.
Here's the longer version:
Sometimes people say that Bayesian statistics are subjective and hopeless. So from time to time it's good to be reminded of my 2017 article with Christian Henning. Beyond subjectivity and objectivity in statistics. There was a lot of good discussion there too. Here is the summary:
Decision-making in statistical data analysis is often justified, criticized, or circumvented using the concepts of objectivity and subjectivity. We argue that the words „objective“ and „subjective“ in statistical discussions are used in a largely unhelpful way, and replace each of them with a broader set of attributes, I suggest replacing objectivity with the following words: transparency, consensus, fairness and Responding to observable realityand subjectivity is replaced by: Be aware of multiple perspectives and context-sensitive. Together with stability, these constitute a set of virtues that are considered useful in discussions of statistical foundations and practice.
The advantage of these reformulations is that the replacement terms do not conflict with each other and give more specific guidance about what statistical science is trying to achieve. Instead of debating whether particular statistical methods are subjective or objective (or normatively debating the relative merits of subjectivity and objectivity in statistical practice), we argue for transparency and multiple perspectives. Desired attributes such as approval can be perceived as complementary goals. We demonstrate the implications of our proposal using recent applications such as pharmacy, electoral voting, and socio-economic stratification. The purpose of this document is to encourage users and developers of statistical methods to use diverse information sources more effectively and to recognize assumptions and goals more openly.