According to the Thai online immigration site, a person with a tourist visa or in TS can apply for a change in visa status at the immigration Bureau downtown. This is to change to a Non-immigrant Visa. (3.3)

So, why is it then that everyone has to do visa runs to change their status? Or am I missing something?

www.immigration.go.th

What you say is correct but it is a fairly recent innovation.

I think you will find that the people doing visa runs at the present time are changing

their status from Visa Exemption to Tourist, which can only be done by applying

for a new visa overseas.

It is possible to change from a 30-day Visa On Arrival Stamp that one gets for free when arriving in Thailand at Bangkok Immigration. Go to the third floor--room 303. You are supposed to have at least 21 days left, but I got mine changed over to a Non Immigrant O Visa with about a week left on my 30 day VOA. However, you need to show a bunch of paperwork and need to have a certain amount of money in the bank. But they let me by with about 3/4 of the money they required after I produced all the other paperwork.

I have also heard that some schools with a bit of pull can change your 30 day VOA to a 3 month Non Immigrant B Visa.

It does not take any pull but a bunch of paperwork. There are a number of categories that can be changed by Immigration and they are listed here under the change visa pop-up.