Why does it seem (perception) as if specialty school/course slots are so few? I left the U.S. Army in 2011. Was this perception false?

It's not a perception, every school has finite seats every year.
And they all cost money.
And the Army gets less money than it needs every year for everything other than payroll.
Which means the Army gets less money than it needs to qualify every Soldier in jobs with required ASIs.
And the Army spends it's finite money on what it needs, not what 1.1 million active and reserve Soldiers want.
You may want wings, but if you're not in or on your way to an airborne unit, the Army doesn't need you to be airborne.
You may want to go to sniper school, but if you're not in one of only three required slots in that BN, the Army doesn't need you to be sniper qualified.

It’s all relative. School seats and availability of school seats to individual units are two different things.

Funding is a major factor and when you combine that with need not all units can pay for training or need the specific training in question.

There are a lot of factors

The "perception" seems to be the view of one individual who wanted some particular school and was unable to get it. That is totally separate from the overall availability of courses.

Course availability is based on school capacity and funding. The Army funds schools to run the number of courses and classes that the Army needs to fill its requirements. Very nearly every seat in every school is based on some position somewhere in the Army needing a soldier who has received that training. Unless the army's needs change unexpectedly there is never any shortage of school seats, but there are also almost no extra seats left over just for someone who wants training they don't specifically need in their current job or in a job they have just been selected for assignment to.

So, having been intimately involved with the schoolhouse side of things, I will say that there is almost never any overall shortage of course slots. The Army's various schools all manage to meet the ARPRINT even when DA and TRADOC impose funding cuts that don't meet the school funding needs.

The Army tries to allocate school slots to different major commands and units based on the number of spaces that unit has which need that particular course. In some cases units don't take advantage of available slots that have been allocated to them. The reasons units fail to use available slots vary. For some schools, the unit has to provide part of the funding. In nearly all schools the unit at least has to cover the TDY and travel costs for their soldiers to attend - the unit may be unwilling to do this. In some cases a unit won't send a soldier to a school he or she needs because they consider that soldier 'mission essential' and aren't willing to lose the soldier for the time of the course. This is generally a short-sighted approach, but most units are short-sighted at least in part because the leaders are only concerned about what happens while they are there. Long-term effects are the next guy's problem. In other cases, a unit might not have anyone qualified to attend a course -- or the individual filling the slot who needs the school might not be qualified for the school. Most schools have requirements for rank, MOS, GT score, and other factors. There are ways to get waivers for these requirements but those take time and effort that many units are unwilling to expend.

When units fail to fill their allocated school slots, those slots become available to some other unit -- but often this happens at short notice which makes it difficult for units to fill those slots unless the unit happens to be located at the same installation as the school. This is one reason that schoolhouse guys sometimes seem to have an advantage getting slots in schools they want to attend.

If a soldier is qualified for a school and is in a unit with a need for that skill and the soldier is in a position that needs the skill, a school slot is almost always available if the unit chooses to send the soldier. But, as noted above, units don't always send soldiers even when the soldier wants the school and is qualified and is in the right job.

On the other hand, there are also a certain number of soldiers throughout the Army who want a particular school simply for promotion points, or for prestige, or even because they think the badge would look great on their uniform. If they happen to be in the right place at the right time, they might manage to attend the school despite it being for all the wrong reasons. If they don't happen to be in the right place at the right time, then they will feel that the Army denied them a chance to attend the school they wanted.