Low voltage filaments can be much more tightly wound to give them greater strength and vibration resistance.

Vehicles started at 6 V, then went to 12 V.
The filaments are 'wrapped' twice to give them enough strength when hot to resist vibration.

Just the vibration of using the bottom shelf in the cabinet above puck lights will make the 120 V bulbs fail very quickly.

24 V is till low voltage.

I would tell your contractor that YOU are the customer.
he appears to have forgotten who is paying him.

It is green, the color of money.

Direct wire advantages (owner's perspective)
1. More flexibility and choices. For example, you can put in fluorescent or halogen and swap it out later for LED with minimal hassle. There is no need to think whether the original setup was 12V or 24V DC and whether the wiring would still be usable for higher output lights which may have a different voltage.
2. For large setups, it is potentially cheaper and there is no need to figure out the wire gauge, # transformers (power supplies).
3. Mechanically secured, does not rely on adhesive (typical for tape).
4. Everything is located under the cabinets. No remote power supply in an accessible location.
5. Better understood by electricians.
6. Higher output than tape lighting and potentially rigid low voltage LED bars.

Low voltage bar advantages
1. Significantly smaller profile than direct wire. Ultra low profile direct wire lights are very expensive and still not as low profile. For example the Unilume.
2. Mechanically fastened vs adhesive for tape lighting.
3. Higher output than tape lighting. In some cases, it can match the output of some direct wire lights.
4. More rugged than tape lighting.

tape lighting advantages
1. Smallest profile of all.
2. Can be water proof (not counting cut ends or connectors).
3. Easily put up, but the adhesive backing may not last.

If you can use fixtures with a 0.75" profile, direct wiring is the cheaper option.