STEM study as productivity?

So I've been mulling it over a bit and I think STEM (Science, Tech, Engineering, Maths) studies should be counted as productivity.

In the most basic sense - the more people in our economy that have training in STEM fields, the more secure we are. More specifically to productivity (the production of new goods) STEM field qualifications could be seen as goods in that they are useful and desired by industry the same way that cars are useful and desired by commuters.

Indeed an individual who has a significant STEM qualification holds a valuable asset that employers will pay a premium to access. The skills are very valuable.

On the other hand... 

The extra pay can be said to go into increasing the price of the goods produced - increasing their value, and so increasing the value of new goods produced. So does that mean we are already counting the use of STEM skills as part of GDP?

GDP also tends to refer to final goods, not components to be used in further production - skill creation could easily fall into that latter category.

So, maybe it shouldn't go into technical analysis under GDP, but does that mean we shouldn't consider it to be productivity at all?

Given that our (UK) government seems to be fine with the idea of supporting companies with direct subsidies, even ones which do not directly contribute to GDP figures, I am going to prefer the idea that STEM education should be included in GDP calculations when considering economic policy, just not when actually measuring GDP.

My reasoning is that when a student studies STEM, they are acquiring an asset that will contribute to GDP in the future. Is this (relevantly) different, from the government's viewpoint, of a bank approving a business loan?

The (UK) government seems very concerned with making sure the banks are willing to lend.
My position is that they should be equally concerned with making sure people are willing to study.