Based on your tone, it sounds like you've already made up your mind. But in the case that you're asking in good faith: the idea behind the mantras is that it is a meaningless sound that preoccupies the language part of your brain without giving it an actual concept to think about. When you repeat and focus on the mantra in this way, the process of meditation feels more like falling asleep (this is what they mean when they say "effortless"). I found this technique far easier and more effective than any other type of mediation. There is no reason you could not look up a mantra (or come up with your own) and practice this technique for free.

TM does charge a fee, which I found reasonable for 90 minutes of 1-on-1 instruction, 4 evenings of classes, and lifetime access to any TM center worldwide. They also have a sliding scale and many opportunities for scholarships. They will tell you that 1-on-1 instruction is integral to the learning process. I did mindfulness meditation for 3 years before switching to TM, and the value I got from the instruction was to simplify my natural tendency to over-complicate these processes (don't think, do). The instruction was absolutely necessary for me, but it very likely would not be necessary for a subset of people.

I understand the inclination to dismiss it as a pyramid scheme or a cult, as the latter was my hangup to trying TM sooner. My opinion: if it is a cult, then it's a bottom-up cult (in that the source of its cult-ness is the obsession of the followers, rather than as the intent from the top). This is very similar to my experience at a Tony Robbins 4 day event; the two days with Robbins were life-changing and _phenomenal_, but the days without him were so bad that I walked out. I tried talking to other attendees about the qualitative difference, but literally nobody was having it, and some of them even got aggressive with me. However off-putting bad sessions and rabid followers may be, it does not change the efficacy, meaning, and power of the experience I had.

TM has a free 90-minute class that I encourage everyone to check out, once we're out of this mess.

The chants might help for people who have certain autoimmune conditions, it's a bit of vocal rehabilitation / voice therapy. Some of the mantras are tongue twisters :)

I know someone whose lupus went into remission after they started chanting, but there was one very particular chant that they found working for them.

That's not a source.