Experienced American English Teacher Confused by Contradictory Info

It is definitely possible to get a job and visa, but it might not be easy.
There is no visa for freelancing, so you need full-time employment.
Since public schools require an educational degree (as a rule), you are limited to employment at private language schools that value experience over certificates.
Unlike in Asia, Germany has only few private language schools (a short Google search showed 20 in and around Berlin) and English (which is taught in all public schools to a satisfactory degree) is not their main business.
To employ you, a school would have to give convincing evidence that there are no available EU citizens that can do the job. For just English teaching, that is very unlikely (at least as long as Britain is still in the EU). So you need some other niche skills that are rare here - and I personally don't think American English will do the trick.
Furthermore, to teach a foreign language (at least to beginners) you need to be able to converse with the students in their home language. This is a requirement of most schools and without sufficient German skills (minimum B1, better B2 level) they might not want to hire you.

I appreciate the response, but could you clarify a bit about there being no freelance visa? Did that change recently? Basically every site I've read about ESL in Germany mentions that ESL teachers are usually considered freelancers, such as this one (specifically about Americans):
https://www.internationalteflacademy.co … in-germany
"6. If you are working freelance (most ESL teachers in Germany are), you cannot begin working until you receive a tax number, which can take 2-6 weeks."

Also I checked some English centers pages and they specify their positions available are for freelancers.

I can find plenty of information about visas for self-employment. Is this something different from what they refer to by freelancing? (Edit: After looking more closely it would appear that the general employment visa is what I'm after, but I'm still confused by this word "freelance" that's going around everywhere.

I guess the next step would be to contact an embassy and wait for a response, but even then there's no guarantee that their answer would be correct in how it applies to me, or that officials in Germany will exactly agree with them when I get there.

All EU citizens and others who have a work permit by other means (e.g. through an EU spouse) can do freelance work. (And yes, basically all ESL teachers are freelancers outside of the formal schooling system, which requires an educational degree as I said).
Please ask the German embassy in your country, as I might be wrong here, but I think there is no work visa purposely for doing freelance work in Germany. ESL teaching is also not on the list of professions with shortages, so don't expect special treatment.

One more note: Freelance teaching isn't well paid at €10-20/hour. If done part-time (which is inevitable as a freelancer), this might not be enough to survive.

Like Beppi mentioned, there are plenty of Americans that already have a residency in Germany that they can apply for the jobs. Often they are married to a German or their spouses are based here in the military. This is why employers are unlikely to go through with the paper work needed to get a non-EU member a visa. Why should they when they already have plenty of people available? And many of these people are not the main bread winners in the family so they might not mind working limited hours for relatively low pay. It gets them out of the house and adds to the family income. One really needs a special sought after skill that is in short supply.  That said, many schools or companies giving training in English will see it as an advantage to have a native speaker.

Theoretically, if you were in Germany without a working visa then you would not likely get controlled if offering private tutoring but you would not likely get hired at any language institute. But this will not afford you any legal basis to stay; on the contrary you would be at risk of getting in trouble for working illegally. Foreign students often offer tutoring in their native language or field of education to make some pocket money but they will be legal on a study visa. 

I looked at the website you posted. It sounds correct in detailing all of the steps needed. The main thing is finding a language school that is willing to hire you. Can they give a letter of intent to hire someone outside the EU that doesn't already have a visa? As far as I know, it takes a lot of paper work for them to justify it on the basis that they were not able to find a suitable EU citizen to do the job. So this is no small detail. It's like giving an explanation of how to win the lottery; one goes to a shop and fills out the forms, one pays and takes the number. Later one checks it against the winning numbers. Then one contacts the office to pick up their winnings… Yeah, the small detail is that one has to pick the correct numbers.

And the website is selling TEFL courses. They have an interest in making their service sound attractive. Such accreditation is probably helpful in getting a job teaching English – in the places it is in demand and one can legally get hired. So, they are correct in saying how it is theoretically done but the probability sounds low.