That is correct.

Moreover, the amplifying function must exist at least in some gates, to restore the logic levels, but there is no need for it to exist in all gates.

For instance, any logic circuit can be implemented using AND gates and/or OR gates made with diodes, which have no gain, i.e. no amplifying function, together with 1-transistor inverters, which provide both the NOT function and the gain needed to restore the logic levels.

The logic functions such as NAND can be expressed in several way using simpler components, which correspond directly with the possible hardware implementations.

Nowadays, the most frequent method of logic implementation is by using parallel and series connections of switches, for which the MOS transistors are well suited.

Another way to express the logic functions is by using the minimum and maximum functions, which correspond directly with diode-based circuits.

All the logic functions can also be expressed using the 2 operations of the binary finite field, addition (a.k.a. XOR) and multiplication (a.k.a. AND).

This does not lead to simpler hardware implementations, but it can simplify some theoretical work, by using algebraic results. Actually this kind of expressing logic is the one that should have been properly named as Boolean logic, as the contribution of George Boole has been precisely replacing "false" and "true" with "0" and "1" and reinterpreting the classic logic operations as arithmetic operations. It is very weird to see in some programming languages a data type named Boolean, whose possible values are "false" and "true", instead of the historically correct Boolean values, "0" and "1", which can also be much more useful in programming than "false" and "true", by simplifying expressions, especially in array operations (which is why array-oriented languages like APL use "0" and "1", not "false" and "true").