Not sure where the disagreement is here. There are X number of jobs in a given industry like the airlines, and then there is a supply of people in the country who want to work those jobs (let's call that number P).
The Bureau of Labor Statistics figure we talked about of (4000) is the X figure change in jobs over the next ten years. It's negative, which means that many less jobs. Obviously if there is a sufficiently high X figure then we will get a pilot shortage, because have not enough Ps to fill the X jobs. Shortage or excess is not reflected in that X figure either, which may be somewhat confusing to some.
You in particular seem to be more concerned with projecting the P figure in ten+ years. If in ten+ years X is smaller than P, then we will have an excess of pilots, and vice-versa for a shortage. However, the BLS says the change is X will be negative so it will be lower to the tune of 4000 jobs. They do not give a P figure specifically.
P is also available currently-known data point, ie. how many pilots there are working currently in the airline industry. It reflects neither shortage nor excess, it is just a report on how many are currently working. There could be tons sitting around collecting unemployment or there could be many more jobs than people to fill them, yet this number would be the same. P is currently around 80,000.
You seem to think that BLS does not consider the X-P equation, but they do even though they do not assign a value to P. They do not come out and say "shortage" or "excess" either, but they do say things like
Pilots seeking jobs at the major airlines will face strong competition because those firms tend to attract many more applicants than the number of job openings. Applicants also will have to compete with furloughed pilots for available jobs.
I am not going to put words in their mouth, but they do project a shrinking number of airline jobs (negative X) while normal growth in any industry is around +11% over a ten+ year period, and they go on to say what's pasted above about competition being stiff at the majors. That's talk of an excess, not a shortage.
A smaller X is a sign there will be an excess of pilots. Without knowing P we cannot say for sure, but I offer (in addition to the words of the BLS) the fact that active ATPs in the US is a stable figure over the last 20 years, and that ATP creation is also fairly steady. Data for that is readily available from the FAA airmen certification statistics pages. Here is a chart of it I made some time ago which I have posted here numerous times.
This chart speaks to the projection of P by showing that no great trend is evidenced by historical ATP certificate data. I appreciate your arguments that we don't really know what P is going to look like in ten or more years, but crying "shortage" without any better data than you have is irresponsible and lacking in support.