So my only conclusion is that they didn't want me to play what was on the page.
Correct.
Standard notation is very flexible, but it often only allows for an approximation of how the piece might be expected to sound. What you may need to do is put in very slight timing and dynamics variations that, if notated accurately, would make the page a hideous mess of tied notes and accent markings.
Try to work out what genre/style the piece is (or what styles it alludes to), and listen to renditions of pieces in that style to work out exactly how the notes are being subtly shifted and accented. Some styles might need you to slightly delay the eighths or sixteenths. Some styles might require you to accent different beats in a way that isn't obvious from the time signature. Some styles might require some sections to be played in quite free time, getting away from the 'grid' implied by the time signature. Or you might have a slightly extended rest between phrases. And so on and so forth.
You're right, of course, that "play with feeling" on its own doesn't give you enough information to tell you what to do. You need to be familiar with the style in order to understand what's expected and appropriate. To an extent, this is where the 'feeling' comes in - if you enjoy and are 'into' a style, you're more likely to understand exactly what timing and accenting variations are right for the style.
But when it comes down to it, it's not really about 'feeling' at all. A good session musician will be able to look at a part that they think is thoroughly boring, and still play it 'with feeling' - because they understand the style.
