The role of depth of a circuit in its hardware implementation
In the zeroth order sense, it is correct that the logic depth and the time to execute the logic would be the same.  There are nuances to this because you need to do something with the result.

What logic is a medium to do work.  In the simplest sense, you have a bounded function with inputs and outputs:  



In most actual systems, you need a way to hold the data, which is why we have latches:



and with the latch, we introduce the clock.  I will now give you two examples of where logical depth, in a system context, is not representative of speed.  

The first assumption that made is "0"s and "1"s are things.  Digital is a special case of analog; however, I personally never run anything at a "1" because it is inefficient for power.  After you are above threshold, you lose transistor gain.  If you are trying to have power performance, you might not make the same choices as for speed performance. You are doing the same electrical work at a high total cost.

The second assumption is that everything is clocked.  In a system context, I often add logic to make things go faster.  Asynchronous systems give you the average speed as a delay, instead of the worst case delay as a synchronous, clocked system.

Another assumption is that logic doesn't have physical space.   Wires are costly even though that they do not add to calculation value.  A good example of this is the S-Box used in AES.  If you look at the "logic", I have often seen hardware implementations that use a lookup table.  This is because it's easy.  If you look at the gate depth alone, without the wires, it also would look to be fast.  When I make the AES S-Box, the circuit implementation looks like:



This circuit has a much larger depth than the lookup table; however, it requires, less power, takes up less area, and is faster than the lookup table implementation even though the logic depth is deeper.