Being in the business of manufacturing both indirect and direct gas-fired heating equipment, my perspective is a little different.
Direct-fired is cheap. That's the bottom line. We sell more of those units than anything else. The construction is simple, pressure losses through the unit are low, and they last forever with a little maintenance.
Indirect-fired is more expensive, both to buy and to operate. The exchangers have a limited life span - we warranty for ten years, some other suppliers won't give you that much. Pressure drops through the unit require more horsepower. There are more components to break down (good for our parts department

).
Both are safe. We've never had anyone injured by a direct-fired unit. However, IAQ is a consideration. If the moisture or traces of CO and NOX can cause problems, indirect is the way to go.
Next, if you are the specifier - stick to your spec. We've lost many jobs because we were spec'ed, bid our own equipment, and someone came along with a low-cost alternative that the contractor liked and the engineer approved....