I have tried using warp:arc lower and warp:rise which was somewhat closer but still cant figure it out how to do that exact style.
2 Answers
That exact style is manually adjusted to work.
You won't find any automated solution to replicate that precisely. Automated methods will only get sort of close to it. Warp, Text on a Path etc. simply won't create similar variations shown in that image. The only want to achieve those is by manually drawing or adjusting the type.
All the variations in individual glyph warping and stroke weights clearly indicate to me all that was done by hand. In addition, that appears to be a wooden sign... and while laser engraving is a possibility.. the inaccuracies, or "wobbles", when comparing the inner and outer paths of the "O"s alone would still indicate the artwork was all manually generated if not drawn onto the wood as a template, then carved/sandblasted out.
Someone drew that type.
While I agree the original looks hand made, you could do something similar with Object > Envelope Distort > Make with Mesh
This isn't automatic and will require manual warping and tweaking of the curve handles until it looks right.
In the rough example below, I used a 1x1 mesh for the middle of the text without the letters A and N, which I added separately and warped with separate 1x1 meshes. With a bit more care and time, I'm sure you could make a better job of it.
