Just watched your video (nice shirt, by the way).

I'll try to answer your question but when Michael comes along he might have a better answer. Edited to add: I was still writing this while Michael responded so see his post for the short and sweet version.

The experiment with Pavlov's dog was
classical conditioning. Clicker training is
operant conditioning.
However, you are right in that teaching the animal to associate a click with a treat
in and of itself is classical conditioning.
In classical conditioning you are working with involuntary responses, so, using Pavlov's dogs as an example, the involuntary response was the salivation (an automatic, natural response to seeing food). The "neutral stimulus" was the bell (which later became the conditioned stimulus), and when paired with the food (the unconditional stimulus) enough times the bell became a "conditional stimulus" at which point the bell itself caused the involuntary response (salivation).
The reason why clicker training is operant (meaning "voluntary") conditioning is because you are not working with involuntary responses. A bird does not automatically have a natural reaction to a targeting stick or trick prop, so the point of operant conditioning is to modify behavior using a type of reinforcement. There are four different types of reinforcement and the one used in clicker training is "positive reinforcement" meaning that you reward desired behavior and thereby increase the frequency of that behavior.
Once the clicker has been established as a conditional stimulus it's used as a "bridge" (or "secondary reinforcer"). A bridge is used to mark the
exact moment the bird does something desirable and
also bridges the gap in time between when the bird does the desired behavior and when it receives a treat.
Once the bird associates the click with a treat you must
always follow the click with a treat or you risk the clicker becoming less effective. There is an exception to this and that is when you are using a ratio schedule of reinforcement (you do not reward every time but rather on a variable or fixed schedule) as opposed to continuous reinforcement (rewarding every time).
If you are just starting out with clicker training you're best starting off with rewarding every single time for
new tricks, and once the bird has a trick down-pat you can phase out the clicker and choose to reward on a variable schedule.
If you still are wondering about how to get your bird to cooperate when it's being less than cooperative I can post about that in a follow-up but hopefully this answers your core question:
You don't want to use the clicker if you're not going to treat because what you have taught the bird through classical conditioning is "the click marks desirable behavior and also means you have a treat incoming." Since the bird is conditioned to know that click means treat that's what you should deliver every time. If you do not pair the clicker with a treat the clicker will lose its meaning because unlike Pavlov's dogs you are not using the clicker to elicit an involuntary response (in Pavlov's case salivation).