Can't really do this in Photoshop. Photoshop's type on a path is as basic as it can be providing no options or refinements.
You may find for Photoshop, using the Warp options located at
Edit > Transform> Warp will yield much better results than type on a path ever will.
Realize that these types of type manipulations on stocks and bods were all done by hand where the craftsman painstakingly made certain every stroke and counter was "just so" to get the effect he/she wanted. To create the same effect, and create it convincingly, you need to design with the same process in mind - carefully and manually.
In my opinion, Photoshop will only ever do a semi-satisfactory job at this type of stuff. Sure you can find a cool font and then use the warp transformations to get it to bend. But it will never look 100% correct due to the limitations of the transformations. Character stroke and counter weights can vary using transformations, note in your sample (and almost any sample you find which is similar) stroke weights do not vary unless there's an intentional reason behind it.
I would do this in Adobe Illustrator and not even attempt it in Photoshop. Using Type on a Path in Illustrator you can set the options to Skew (
Type > Type on a Path > Type on a Path Options...) which will maintain the verticals of the type and only alter the horizontals.
From that point you could create outlines and refine. Or simply copy/paste to Photoshop as a smart object if you must use Photoshop.
This is Illustrator with type on a path set to skew using a decorative font and an additional, offset, fill in the background to create the shadow.