How do logic chips (74LS, HC, ...) provide output=Vcc/0V when all standalone diodes and transistors suffer voltage drops?
You are taking knowledge about one kind of circuit and applying it to all kinds of logic chips.

Logic devices that use bipolar transistors, such as TTL, are limited by the \$V_{cesat}\$ of the transistors, so they can't pull the output very close to ground or to Vcc (unless the current is near zero).

MOS transistors work much differently, and they can in fact drive the output voltage very close to Vdd or ground for low values of output current.

You should compare the \$V_{OL}\$ and \$V_{OH}\$ specifications of different kinds of logic chips to get an idea of how they really behave.