About as far as you can take it with PDFs out of the box is just "clicking" the button via js to initiate the generation of a pdf as if you / a user had done it. Actually controlling the contents of the pdf or where it goes is not something that at least I know how to do. 

Not to mention, the PDF documents generated from the built in PDF creator, while better than nothing, leave a lot to be desired as it's essentially just as if you had printed the HTML page which isn't always desired (because it lacks all sort of formatting and other such "branding" typical of what most businesses want). 

I personally went down the route of using a third party pdf gen javascript library called PDFMake (http://pdfmake.org/#/) which can be used to generate pdfs however you may desire but requires some work to get setup in such a way that it's making them to your exact specifications. And while I am not currently doing something like generating a pdf to a stream and then automagically uploading it into sharepoint at the click of a button, I believe that it *can* be done in a straight forward way using the SharePoint REST API (or better yet, something like SPServices https://github.com/sympmarc/SPServices).

The only other option I could think of would be to go down the Nintex DocGen route https://www.nintex.com/process-automation/document-generation/

So yeah. It's a "build it yourself" or "find an existing product that works and buy it" kinda situation! 

Here is another idea to consider.  Use the Document Generation workflow steps to do the conversion in the workflow instead of the form.   

I use a product called SharePoint Convert to PDF by Muhimbi: https://www.muhimbi.com/products/pdf-converter-for-sharepoint/

It integrates with Nintex Workflow perfectly and generates nice looking PDFs.