担ぐ and 持つ: carrying on the shoulder and holding more broadly
Compare shoulder carrying with holding a box. Weight or distance alone does not identify the shoulder posture; also preserve broader possession uses.
Carry the load on the shoulder
担ぐ, read かつぐ, means carry on the shoulder in this luggage scene. 重い荷物を肩に担いで階段を上った。 says that someone carried the heavy load on the shoulder and went up the stairs.
肩に explicitly gives the shoulder posture. Going up the stairs is completed; the posture is not inferred from the weight alone.
Suggest holding and carrying with 持つ
持つ, read もつ, means hold or have. 重い箱は二人で持って運ぼう。 suggests that two people lift and carry the heavy box together.
二人で gives the number of participants and 運ぼう is a suggestion. It does not report completed carrying or specify carrying on the shoulder.
Read uses beyond physical objects
彼女は自分の意見をしっかり持っている。 says that she firmly holds her own opinions. This use differs from holding an object in the hands.
仲間は彼を次の代表に担ごうとしている。 says that the colleagues are trying to put him forward as the next representative. It does not report that he has already been selected.
Does 持つ for a heavy load alone establish the shoulder posture expressed by 担ぐ?
Show the answer and why
No. Distinguish broader holding from shoulder carrying. Check an accompanying position expression such as 肩に for the posture.