Pat Thompson has done research on students' and teachers' understandings of functions and variation. (He's a co-author on the article mentioned in this answer, though he's not listed for some reason in the link you gave in your comment.)
In a short 2013 article, Thompson calls the statement $C=f(x)=3x+2$ a misuse of notation that can lead to "[confusing] the idea of function and variable." In the same article, he relates this to when students see a function definition like $V(u)=u(13.76-2u)(16.42-2u)$ and think that the name of the function is $V(u)$, i.e. $V(u)$ is simply a label for the right-hand side. He also makes the following claim about function notation that agrees with the sentiment of your answer to another question.
A primary source of students’ difficulty with function notation is that they only see it where “𝑦” could be used just as well. The textbook says they must use 𝑓(𝑥) when there really is no need for it. They rarely see function notation used in settings where using it actually enables them to do things that they otherwise could not.
Yoon and Thompson (2020) offers several more references on students' and teachers' difficulties with function notation.
Sajka’s (2003) case study involved a 45-minute interview with one high school student, Kasia, who had seen function notation for three years. [The study] showed Kasia moving from thinking "𝑓" means “the beginning of a function formula” to thinking of “𝑓(𝑥)” as serving the same role as “𝑦”... Sajka’s report that Kasia understood “𝑓(𝑥)” as the start of a function formula is in line with Thompson’s (1994) report of students thinking a function definition consists of the “𝑓(𝑥)” on the left side, the symbol “=”, and an algebraic expression on the right side. Thompson (1994) also pointed to this way of thinking as behind the common mistake of mismatched letter on the left and letter on the right, such as 𝑓(𝑥) = 𝑛(𝑛 + 1)(𝑛 + 2).
Musgrave and Thompson (2014) and Thompson and Milner (2019) shifted focus from students’ to teachers’ understanding of function notation. Musgrave and Thompson (2014) and Thompson and Milner (2019) found that, for many teachers, function notation served as a label or a name for the defining formula rather than a representation of one quantity’s values in relation to another quantity’s values. While students meaning of function notation as a label might be a root of so many reports of students’ difficulties with it (e.g., Carlson, 1998; Dreyfus & Eisenberg, 1982; Vinner & Dreyfus, 1989), it is important to investigate the possibility that teachers hold similar meanings.
Yoon and Thompson's study asked teachers in the US and in South Korea to respond to tasks involving function notation, and they give many examples of what I think aligns with your idea of "blurring between functions and variables."
A large majority of US teachers seemed to think that “𝑣” in 𝑐(𝑣) is a part of the function name and they were therefore free to use other letters in the function’s defining rule. Thompson (1994, 2013b) suggests a reason for the teachers who used 𝑠 or 𝑡 at least one blank in Level 2 and Level 1. Teachers thought of function notation as a four-character symbol that is used in place of the letter “𝑦” (Thompson, 2013b). Teachers who filled the blanks with 𝑠 or 𝑡 might consider “𝑤(𝑡)” as one symbol because they thought they could replace “𝑤(𝑡)” with “𝑦”.
Teacher 1’s response... tells us that she thought 𝑞(𝑠) was one inseparable symbol instead of thinking 𝑞 is a function’s name and 𝑠 is an independent variable... Although she said 𝑞 is a function of 𝑠, she thought 𝑞 always accompanies with 𝑠, and viewed 𝑞(𝑠) as one entity.
Teacher 1 first wrote 𝐴 = π𝑟2 , and then added (𝑡) after 𝐴. Her final answer was 𝐴(𝑡) = π𝑟2 . She used function notation only to represent the area on
the left hand side and used 𝑡 in the function notation and r in the defining rule. Her statement “when I am using function notation I am thinking 𝐴(𝑡)” and “I was just noting the area of the circle which was what I was starting” is consistent with our hypothesis that teachers who used function notation only to represent the area think of 𝐴(𝑡) as one symbol, and 𝐴(𝑡) is a label for the formula on the defining rule.
One of the study's interesting conclusions is that
SK data shows that US teachers’ difficulties when reasoning with function notation are not due to epistemological obstacles to understanding meanings of function notation. The SK data suggests that US teachers’ problematic responses are a systemic aspect of mathematics education in the U.S. Thus, it is plausible that US teachers convey problematic meanings to students unintentionally.
In Thompson and Carlson (2017), the authors discuss the concepts that you call "function" and "function of" and argue that many student difficulties stem from an overemphasis in school mathematics on the modern, set-theoretic definition of function and on a static conception of variables. Instead, they argue that there should be more focus on the notion of what you call in this MO question "variable quantities."
We argue that ideas of continuous variation and continuous covariation are epistemologically necessary for students and teachers to develop useful and robust conceptions of functions. Put another way, we argue that variational and covariational reasoning are fundamental to students’ mathematical development. We ground this claim in research that highlights difficulties students experience regarding function relationships by not having the ability to reason variationally or covariationally and in research that shows productive shifts in teachers’ and students’ conceptions and uses of function that result from reasoning covariationally.
They review U.S. and Japanese school mathematics textbooks and find that
Our cursory review of 17 U.S. secondary precalculus level textbooks ranging from algebra 1 to precalculus, revealed, consistent with Cooney and Wilson’s (1993) textbook review, that all the textbooks used a correspondence definition of function. The research we have cited in this chapter further supports the fact that U.S. curriculum and instruction are failing to develop students’ quantitative and covariational reasoning abilities, contributing to many weaknesses in students’ conceptions of fundamental mathematical ideas, such as variable, function, and rate of change, that are essential for understanding calculus and modeling dynamically changing phenomena in the sciences and engineering.
The Japanese primary mathematics texts have a clear, coherent focus on having students think about quantities whose values vary and about ways that quantities’ values vary together. By high school, Japanese texts’ authors presume that students think with images of variation and covariation and rely on this assumption as a matter of practice. In contrast to Japanese textbooks, many popular U.S. textbooks do not emphasize or support students in conceptualizing quantities and viewing function formulas and graphs as representing how two varying quantities change together. The idea of variable often is presented as representing a single unknown value.
Based on this, I think there is substantial support for the claim that the "function" vs. "function of" issue and the "function" vs. "variable" issue do indeed affect student understanding, and that teachers' actions contribute to the confusion.