The pressure generated at Top Dead Center on the Power stroke is very high. A "taller" gasket will have a larger cross-sectional area exposed to this pressure, so high "pounds/sq inch" grows proportionately as the area increases.
Even the best Multi Layer Steel head gaskets are not designed to be part of the cylinder, just to seal the head to the block. It's a gasket, not a spacer. In addition, it will be impossible to achieve proper head-bolt torque with a tall "squishy" gasket.
With a pure copper gasket, you might be able to approach 100 thousandths of an inch total gasket thickness without failure, and proper head bolt torque. Such a gasket would likely be custom and cost more than a new set of heads.
Your mechanic was right to advise against it. (Although many have tried... even such folly as using TWO gaskets to double the height. This doesn't end well in any story where one admits the truth...
On edit: Also keep in mind that you can't just "resurface" the warp out of an Overhead Cam head. The entire head is warped, not just the surface that interfaces the block. A proper head restoration will need a cam line bore and bearings. I've had some luck with aluminum heads, carefully measured and shimmed between clamped steel jig plates, in my kitchen oven as high as it would go for five hours. Cam bore alignment was restored, and only a .010 cut was required. My wife (now ex), however, was NOT amused...