Is there a reason the 月 radical occurs many times in body part names? It's possible I'm reading too much into this, but there seems to be a lot of body parts which contain the 月 radical:


  腿 (tuǐ) = leg
  脚 (jiǎo) = foot
  腰 (yāo) = waist (also 腰部 (yāobù))
  脸 (liǎn) = face
  胸 (xiōng) = chest
  股 (gǔ) = thigh (also 大腿 (dàtuǐ))
  胳膊 (gēbó) = arm
  脂肪 (zhīfáng) = body fat
  肝脏 (gānzàng) = liver
  膀胱 (pángguāng) = bladder
  脖子 (bózi) = neck
  肚子 (dùzi) = abdomen
  脑子 (nǎozi) = brain
  腋下 (yèxià) = armpit
  膝盖 (xīgài) = knee
  肌肉 (jīròu) = muscle
  手腕 (shǒuwàn) = wrist
  肩膀 (jiānbǎng) = shoulder
  屁股 (pìgu) = bottom
  皮肤 (pífū) = skin
  手肘 (shǒuzhǒu) = elbow


Also: 脉, 胆, 肠, 腑, 脬, 腱.  You get the idea; there's lots.

Question: Is there a reason the 月 radical occurs many times in body part names?

There's others that don't have it, e.g., 头 (tóu) = head, 眼睛 (yǎnjing) = eyes, 耳朵 (ěrduo) = ear.  Nevertheless, 月 does seem surprisingly common among body parts.
Similar questions have been asked in both Chinese and Japanese StackExchange before. In summary, what appears as 月 in printed text is actually the graphical merger between ⺼ (simplified form of 肉 (meat; flesh)) and 月 (moon). Traditionally, you wouldn't handwrite characters with the ⺼ radical with two horizontal lines as the inner two strokes; they would be diagonal.

The similarity in appearance is normally not a source of confusion, as the vast majority of characters which contain a ⺼ component does not have ⺼ appearing on the right, and the vast majority of characters which have a proper 月 component appears on the right, with some rare exceptions (see this).

As a final note, if you wish to understand why characters are written a certain way, looking at the Simplified Chinese form is generally not helpful. 头 has undergone a very contrived simplification from the traditional 頭 (see this excellent answer on Zhihu), and like 頭, characters to do with 'head' may be written with the 頁 component instead of ⺼ (e.g. 頸, neck).