Of course not every market is a large efficient market. If you're talking about a price-regulated market, by all means, discuss shortages. Another good situation where discussing a shortage would be appropriate is markets that have very or perfectly inelastic supply over a relevant time period. Say, a vaccine that costs $1 but that takes 6 months to produce due to the biology involved, combined with a major epidemic in the relevant disease. It's fair to call that a shortage.
Constraining supply increases salaries precisely because it takes supply so long to incorporate information.
On top of that, supply and demand is a pretty leaky abstraction, I don't expect it to be very accurate in such a nonideal market.
For any given skill there is a labor pool with some participation rate, of people who currently have that skill, such that it is accurate to say that the market has a shortage.
Answers to the shortage must solve for either the size of the labor pool, the rate of participation, or the amount of people who have that skill.
Those answers vacillate between importing labor to solve the problem now, or forcing the market to educate people by blocking imported labor.
The problem with importing labor is that it reduces market incentives to educate people. The problem with not importing people is that it means you'll have a labor shortage until the market adjusts and people re-educate.
Easy answers, on either side, are snake oil.
"Shortage at the market price" seems to be a somewhat incoherent concept. The market price is the price point at which the supply and demand curves intersect. Other than a discontinuity in either supply or demand such that the curves pass without intersecting (which makes the market price ill-defined), demand equals supply at the market price.
We're pretty comfortable talking about shortages in consumer markets when we would buy things at the stated price, but can't because it is sold out (eg Pixel phone shortages), so I'm not sure why we can't say a similar thing about labour.
That can just be a current price below the equilibrium market price; I'd argue that a shortage is a specialized instance of that resulting from a transitory decrease from long-term supply (or increase from long-term demand) that is expected to revert to normal.
But if you need annual supplements of labor for the same field over a period of decades, that's not a labor shortage, its price being artificially suppressed (and inhibiting market signals that would develop domestic supply faster.)
Theoretically, couldn't it also be that demand is consistently growing faster than anticipated (even after whatever corrections are made upon discovery that it previously grew faster than anticipated)?
I'm pretty sure the word shortage already implies that there is a shortage at the market price, not at an arbitrary price.