Approximate transfer function expression of this circuit
When you want to study a circuit and see how it could reduce to a simpler form when neglecting some of its element values, its is important to express the transfer function in a low-entropy form, a term forged by Dr. Middlebrook when teaching his extra-element theorem or EET (https://en.wikipedia.org/wiki/Extra_element_theorem). A low-entropy form, is an expression in which you see how elements interact with each other, for instance trough series-parallel arrangements. You can also see a leading term which tells you if there is gain or attenuation at a certain frequency point (0 or infinity for instance). So in your example I can show for the sake of illustration how the EET can get you to a simple expression without using node analysis at all. First, select what is your extra element, the one without which all is simplified. I will work with resistances \$R\$ rather than \$Z\$ and will pick \$R_3\$ as my element. I have the choice to make it an infinite- or zero-valued element to determine my reference transfer function. I choose to make it an infinite value and I thus re-draw the circuit without \$R_3\$ to compute the reference transfer function \$H_{ref}\$. By looking at the below sketch, you immediately see that it is

\$H_{ref}=\frac{R_4}{R_4+R_2+R_1}\$



Then, I need to determine the resistance \$R_d\$ when the excitation voltage \$V_{in}\$ is reduced to 0 V (replace the source by a short circuit). When running this simple exercise, I obtain

\$R_d=R_1||(R_2+R_4)\$

The final step is obtained when I determine the resistance \$R_n\$ seen from \$R_3\$'s terminals when the response is nulled (\$V_{out}=0\;V\$) and the excitation is back in place. If I do this simple exercise - designated as a null double injection or NDI - I can see that \$R_n=0\;\Omega\$ since I have 0 V across the current test generator. What is even cooler is that I can use SPICE to check all these intermediate steps and verify (and correct) my results.



I'm done and can apply the EET expression which is

\$H=H_{ref}\frac{1+\frac{R_n}{R_3}}{1+\frac{R_d}{R_3}}=\frac{R_4}{R_4+R_2+R_1}\frac{1}{(1+\frac{R_1||(R_2+R_4)}{R_3})}\$

With the series-parallel arrangement, you can see the impact of each element and check how neglecting its value could impact the transfer function. The below Mathcad file verifies all these results



Of course, the EET is a kind of an overkill here -"only way to feel the noise..." : ), but it shows the basic principle of the Fast Analytical Circuits Techniques or FACTs (see http://cbasso.pagesperso-orange.fr/Downloads/PPTs/Chris%20Basso%20APEC%20seminar%202016.pdf)

The Thévenin approach kindly sugested by jonk is already a first step towards FACTs because it breaks and simplifies the circuit, illustrating the "divide and conquer" theory at work with FACTs.