The answer is "absolutely true, depending on what you're training for."
That is, if you're studying a classical Asian martial art to learn the history, tradition, and spiritual aspects, the speed bag isn't going to do much for your tiger-fork forms or your iaido technique. Oh, it'll do a little, because it's all good fitness training, but it isn't going to help your form or technique.
However, if you're working on self-defense or tournament sparring, which have a high need for speed, timing, and accuracy, boxing is an excellent addition to your martial-arts regimen. We've incorporated boxing training and techniques into our competition training since the 1970s.
As for boxing being the BASIS for a lot of martial arts, that's not the case, at least historically speaking. Asian martial arts traditionally came from India to China when Bodhidarma brought Buddhism into China, founded the Shaolin Temple, and was appalled that the Chinese monks were in such poor physical condition that he taught them the physical training that became gongfu or wushu. This martial art in turn migrated to the Ryukyu Islands (Okinawa) and to Korea; the Okinawan version was adopted by the Japanese and combined with jujutsu, the traditional hand-to-hand Japanese fighting method (which stressed grappling and joint-lock techniques).
Asian martial arts were first widely introduced to the West in the late 1950s, but really began to take off in the 1970s with the popularity of Bruce Lee's movies and the TV show "Kung Fu" with David ("Kill Bill") Carradine. My family was at the heart of this "second wave" of martial arts, and did a lot to get the practicioners of various styles working together to learn from each other. It was an exciting and creative time to be in martial arts, training with Okinawan, Chinese and Korean stylists and incorporating my father's extensive amateur boxing background into a different way of teaching and applying martial arts.
When "full-contact" and "kickboxing" matches first became popular in the late Seventies, it was an eye-opener to many traditionalists that a good boxer could more than hold his own with a good Asian-style martial artist in the ring. It's that whole speed, timing and accuracy thing -- and since then, Western style boxing training (speed bag, sparring, moving target training, and more) have been added to many martial-arts schools, especially if they have a strong interest in tournament fighting.
My favorite quote on the subject comes from Miyamoto Musashi, Japan's most famous swordsman and author of "The Book of Five Rings" (Go Rin No Sho), the best book ever written about the philosophy of Japanese martial arts and the psychology of combat:
"Anyway, cutting down the enemy is the Way of strategy, and there is no need for many refinements of it."