Success!  I had to install/use the following tools:
npm install html-pdf-chrome --save-dev
npm install pdfreader --save-dev
html-pdf-chrome is used to magically call Chrome to convert some given HTML to a PDF in the manner that Chrome would normally use to print.  pdfreader is a package that reads said PDF and then provides the text inside of it.
After browsing to the page I want to print using webdriver, I can call:
this.When(/^I print the page to a PDF named "([^"]*)"$/,
  async function(outputFilename) {
    console.log("Getting the html...");
    let sourceHTML = await browser.getSource();
    console.log("Printing the html using Chrome...");
    let pdf = await HtmlPdf.create(sourceHTML);
    console.log("Saving the PDF to " + outputFilename + "...");
    await pdf.toFile(path.join(DEFAULT_PRINT_PATH, outputFilename));
  });
Then, to get the text in the PDF, I call this function:
function readPdfText(filename) {
  return new Promise((resolve, reject) => {
    let pdfText = "";
    new pdfReader.PdfReader().parseFileItems(path.join(DEFAULT_PRINT_PATH, filename), function(err, item){
      if (err){
        console.log("Error received on parsing PDF: " + err, err.stack);
        reject(err);
      }
      else if (!item) {
        resolve(pdfText);
      }
      else if (item.text) {
        if(item.text.trim() === ":") {
          pdfText += item.text;
        } else {
          pdfText += "\n" + item.text;
        }
      }
    });
  });
}