Simply put, this push-up, calling it a lie, is a bit nonsense. It just can't be used to effectively increase the girth, but it is good to enhance the endurance and separation of local muscles.
If you want to gain muscle, to be honest, it may be of little use to practice push-ups and pull-ups by ordinary exercises. That big brother was right when he said he wanted heavy equipment. Training, plus enough rest and nutrition, can grow. As for the coach asking you to buy classes, that's what people do for a living, so I won't say much.
If you don't want to buy a class, at least you should read some materials to study, and then find a few bosses to guide you to get started. Don't think it's easy, just watch the video. Many movements look the same, but the strength part changes slightly, or the rising and falling time changes, so the practice is different.
Add: Cudgel didn't say people lied to you, just said to be clear about your goals. To increase muscle circumference, heavy training is obviously essential. But it is useless to say that push-ups are also suspected of being taken out of context.