Most of the time I have my Marshal train troops, my Steward collect taxes, and my Court Chaplin research Cultural Technology. Except I don't actually know what that Technology Spread effect is doing precisely.
When a council member improves the technology spread rate, does that improve how fast the county accumulates tech to the next tech levels, or does that make the county spread its increased levels to nearby counties that have lower tech levels, or what?
The CK2 Wiki has an excellent and up-to-date article explaining technology. It has this to say about technology spread:
Technology spreads between provinces that border each other, as well as between provinces in the demense of the same ruler even if they do not border each other. The rate of growth depends on the degree of the technology lead in the more advanced province. The Spymaster's Study Technology mission also lets technology spread from the target province. The other advisors' research missions also increase the speed of spreading, as do the School and University buildings.
So by sending your Court Chaplain (or Steward, or Marshal) to research tech in your capital, you increase the rate at which the capital increases in cultural (or economic, or military) technology and therefore increase its lead over provinces neighbouring your capital, which in turn increases the rate of spread of tech from your capital to its neighbouring provinces and non-neighbouring provinces in your personal demesne.
This is a good thing! Your capital is where tech will increase the fastest, so you want to encourage those advances to spread to other provinces in your realm. If your capital borders directly on provinces outside your realm (for example, Krakow in Poland borders Rus provinces and Hungarian provinces), either accept that you're helping your rivals somewhat, or move your capital to a more central province.
If you want a particular province to catch up faster instead, sending your councillors there will multiply the effect of spread from its neighbours.
