You're right that Shiny needs a server. (Not to mention the user discipline to stop playing with the variations and think critically about the problem. No rap on Shiny; inevitable GUI feature).
Think of Beamer as page at a time. Think of regular LaTeX as basically a typesetting/formatter for text and graphics in the style you might see in an academic journal. Pdf is the output of a LaTeX or pdf file.
It's a standard feature in RStudio Rmarkdown, which has a variety of LaTeX templates that can be installed. The "tufte" still is very popular, but even the default template allows you to output to pdf with inserting results of code as a "chunk" or inline.
Donald Knuth wrote the precursor to LaTeX when academic publishing changed over from hot lead type to phototype printing. He was unsatisfied with the rendering of equations, so he wrote a typesetting language. The basics aren't exactly simple, but able to be mastered. The fine grained, fresh off the press perfection is a calling that can totally displace your concern with the underlying content if you are not careful.