direct answers
Can someone check my work and let me know of any errors?
I think your equations are fine. I don't see any errors.
let me know if there's any better way to format this.
I think whatever works for you is better. I don't see any obvious choice, one way or another. Some will prefer one, others another.
I'm trying to find a more intuitive equation for wye to delta
formations
I can't say what's more intuitive for you. And if this is just a question about which algebraic form "looks better" then I'm stuck there. But there is this page that you might skim.
I can add these thoughts:
\$\Delta\$ and Y KCL
Here's your diagram with \$N_{_\text{Y}}\$ and a few currents added to it:
In the following, I'll be using conductances for everything. So \$G_1=\frac1{R_1}\$, etc.)
From KCL we can make the following initial observation about the Y:
$$\begin{align*}
V_{_\text{Y}}\cdot G_1+V_{_\text{Y}}\cdot G_2+V_{_\text{Y}}\cdot G_3 &=V_1\cdot G_1+V_2\cdot G_2+V_3\cdot G_3
\\\\
\therefore\quad V_{_\text{Y}}&=\frac{V_1\cdot G_1+V_2\cdot G_2+V_3\cdot G_3}{G_1+ G_2+G_3}
\end{align*}$$
The net node currents given in the above diagram for each configuration is:
$$\begin{array}{lcccc}
&\textbf{Y}&&&\Delta\\\\
I_1:&\left(V_1-V_{_\text{Y}}\right)\cdot G_1 &&& \left(V_1-V_3\right)\cdot G_{_\text{B}}+\left(V_1-V_2\right)\cdot G_{_\text{C}}
\\\\
I_2:&\left(V_2-V_{_\text{Y}}\right)\cdot G_2 &&& \left(V_2-V_1\right)\cdot G_{_\text{C}}+\left(V_2-V_3\right)\cdot G_{_\text{A}}
\\\\
I_3:&\left(V_3-V_{_\text{Y}}\right)\cdot G_3 &&& \left(V_3-V_2\right)\cdot G_{_\text{A}}+\left(V_3-V_1\right)\cdot G_{_\text{B}}
\end{array}$$
But \$V_{_\text{Y}}\$ can be expanded and then the above can be re-arranged as follows:
$$\begin{array}{lcccc}
&\textbf{Y}&&&\Delta\\\\
I_1:&\frac{\left(V_1\,\left[G_2+G_3\right]-V_2\:G_2-V_3\:G_3\right)\, G_1}{G_1+G_2+G_3} &&& V_1\cdot\left(G_{_\text{B}}+G_{_\text{C}}\right)-V_2\cdot G_{_\text{C}}-V_3\cdot G_{_\text{B}}
\\\\
I_2:&\frac{\left(V_2\,\left[G_3+G_1\right]-V_3\:G_3-V_1\:G_1\right)\, G_2}{G_1+G_2+G_3} &&& V_2\cdot\left(G_{_\text{C}}+G_{_\text{A}}\right)-V_3\cdot G_{_\text{A}}-V_1\cdot G_{_\text{C}}
\\\\
I_3:&\frac{\left(V_3\,\left[G_1+G_2\right]-V_1\:G_1-V_2\:G_2\right)\, G_3}{G_1+G_2+G_3} &&& V_3\cdot\left(G_{_\text{A}}+G_{_\text{B}}\right)-V_1\cdot G_{_\text{B}}-V_2\cdot G_{_\text{A}}
\end{array}$$
You can now directly compare the left and right columns. By inspection, the following must be true:
$$\begin{align*}
G_{_\text{A}}&=\frac{G_2\,G_3}{G_1+G_2+G_3}
\\\\
G_{_\text{B}}&=\frac{G_3\,G_1}{G_1+G_2+G_3}
\\\\
G_{_\text{C}}&=\frac{G_1\,G_2}{G_1+G_2+G_3}
\end{align*}$$
expanding the topic further
There's a really nice selection of historical papers on the topic, from a modern 2022 perspective, that can be found in the first few paragraphs of A degree preserving delta wye transformation with applications to 6-regular graphs and Feynman periods, by Shannon Jeffries and Karen Yeats, July 29, 2022. I recommend at least skimming the first page there just to get a good list of relevant literature on the topic. (And Karen is the contacting author, if you've a mind for that.)
They cite perhaps the earliest paper on the topic as "Equivalence of triangles and three-pointed stars in conducting networks," Electrical World and Engineer, 34:413–414, 1899 by A. E. Kennelly. (I just obtained a public domain copy.)
Another important paper is "Polyeder und Raumeinteilungen", Enzyklopadie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, 1916 by E. Steinitz.
The above authors are, of course, no longer with us.
Perhaps one of the better living experts on this topic would be Isidoro Gitler. I think he is currently at the "Center for Research and Advanced Studies of the National Polytechnic Institute". His doctoral thesis, at the University of Waterloo in 1991, was "Delta-Wye-Delta Transformations: Algorithms and Applications." He then published "On topological spin models and generalized ∆–Y transformations" in 2002, followed by "On terminal delta-wye reducibility of planar graphs", 2011.
If you are looking for important insights, I'd strongly recommend making contact. Your questions would be squarely and directly related to his work and publications. I expect you'd get a favorable response to queries.
