When China was made P5 member, it had neither the influence nor power projection ability far from its shores.
it did though. while it did not project hard power very far, there were two mitigating factors, one is that china itself exists in a very important part of the world, between us occupied SK, Japan and the soviet union itself. just existing there and having a large military made it important on the world stage, especially after showing itself as capable of fighting after the Korean war. secondly, while it did not project its own power, it could project hard power via its connections, for example arming rebels in africa and selling missiles to the middle east (and this isnt a "could" or "should" but that they actually did), so while china could not say, show up with two CBGs to africa, it could make a lot of AKs and RPGs suddenly appear in places. moreover it was a independent large power free from soviet or american influence(although less capable than those two) and anyone that couldn't work with the us or the soviets could turn to china.
Premier zhou enlai toured the african continent in the 60s. china had strong relations with Zimbabwe and supplied them with weapons along with countries like sudan and egypt and militia/irregular forces like the FNLA.
Remember this was before Mao died and china started the period of non-inference its in now. back then china was very active politically around the world and support many independence movements and foster relationships with many third world nations. you just don't hear too much about it because it's since stopped and US/Soviets did even more back then.
such influence is shown by the very fact that mainland china even got the votes required for its UNSC seat through the general assembly in the first place. Africa and east europe was instrumental in this vote and in fact it was them that had to bring up the issue to the assembly as the PRC could not due to not being a member.
this vote was extremely successful. to give you an idea, the US initially wanted none of it, but when it was clear china had overwhelming support the us started bargaining, including making PRC a member but also keeping RoC as a member of the P5, then trying to torpedo the vote by stating that it should need a supermajority, then making the PRC a permanent member but keeping RoC as a regular member, then finally when realizing that nothing was gonna go their way, they could only vote against it, but the resolution passed with overwhelming support and kicked the RoC out.
That is world wide influence in action right there, even more so when it was against the wishes of the US.