1930s stone / shingle house with slate roof
Hardie board, if properly installed, does not look synthetic. If you paint it, it will look just like wood clapboard. I for one favor the smooth board as opposed to the board with the wood-like texture which I think ironically makes it look instantly fake. We added an addition to our previous house, which was a 1934 wood clapboard bungalow and we did the addition in Hardie board. It was impossible to tell what was wood and what was Hardie when it was finished.
You contractor's main concern is that when you pull old shingles off it's very hard, and often impossible, to do so in a way that renders them reuseable. Shingles are by their very nature a labor intensive installation and so that is something to consider, and your contractor is probably experienced with Hardie board and has confidence in the material and his ability to install it properly. He is certainly correct that it will be easier to do the insulation and window work with the shingles off. The Hardie board is for all intents and purposes permanent so from a cost and value perspective it is the clear winner.
All that said, if you really like the look of the shingles, and you're willing to pay the price to keep them, go for it. If well cared for they last a looooong time.