Several downsides that I can think of:-
Moisture sensitivity; probably ok if fully encapsulated but this can fail due to element flexure.
Charge amplifier needed - at least on the Kistler ones I've used. Set zero and zero drift can be an issue. Linearity can be an issue.
The piezo element by definition measures force applied upon it without deforming - so mechanical element can change characteristic over time.
All kinds of issues - I've done serious battle with most of these technologies when testing oil drilling tech..
Piezo's great for ultra high speed/high resolution (with suitable DAQ), and can be sensitive enough to detect a breath from ten metres away (tried it - amazed) - insane turndown ratios possible, but none of the examples I have met would survive a Landyzone users tender ministrations for too long. Things may well have changed of course in the seventeen years (arg) since I was doing this in anger!
LVDT & Hall effect have the advantage of no physical connection necessary between the static and the live element.
I'm seriously thinking (!) of creating a DIY LVDT (hand-wound, built with pipe and nails) and 4x PID loop (Cheap off ebay) replacement EAS controller - this would be less difficult than you might think....
I'm sane, honest......