Does being a CPL hold some valuable weight when taking care of your soldiers, for example at the different S shops?

It 100% does. In my experience, fellow NCOs give me more respect and naturally hold my input in higher regard than a SPC. Many places and shops require NCO escort for certain tasks, which you are then capable of as a CPL. Additionally, you have more responsibility for the conduct, performance, and well being of your soldiers, as a SGT would, therefore the leadership gives you the latitude to carry out your NCO duties, just as a SGT would. The biggest thing, though, is differentiating yourself. A lot of new CPLs are awkward about requiring their troops to follow the customs and courtesies for their new NCO rank and so they get treated like a SPC in charge because that is all they require of their troops. Make the distinction from the beginning, carry yourself like an NCO and there is no reason you should not be treated as one. SPC is automatic, the stripes are earned.

It would for me... I had a CPL who worked for me and he was outstanding. Certain officers didn't get it, but the NCOs did.

AR 600-8-19 Para. 7-11 c. A SPC will be appointed to CPL when the Soldier is assigned and works in a SGT position. The position must be the Soldier’s PMOS or CPMOS. This includes SPC who cannot otherwise be promoted when they do not meet the TIS requirement, or were not considered for promotion.

This means you will fill a SGT position. Even though your Rank is CPL you have the duties and responsibilities of a SGT. Therefore, you must be just as much of a leader as any other SGT in your unit.

Go ahead and start looking at DA PAM 600-25 and FM 7-22.7 for insight into the NCO Corps.