You are setting yourself up for multiple post-docs if you want to eventually run your own lab. But really, you'll probably never get that far.
This depends on a lot of things. Is the OP interested in switching fields and go into more clinically oriented research? Or even basic translational research? I find that certain funding mechanisms attract a very different crop of applicants compared to the peers of familiarity at top 10, HHMI, pedigreed cohort. My sense is that at that point it's just the sheer number of grants you submit that determines your success... I feel like people barely look at your biosketch, but of course at this point I don't even know anyone knows what's what anymore.
Let me give you an example. For the MD Ks (K08 and K23), a lot of applicants only have MDs and 2-4 years of postdoc. Yet it has a 30-40% success rate. This is nationwide. I can tell you the vast vast majority of mid tier R1 medical centers junior faculty with a K don't have the kind of profile you think they would have. On the other hand, certain foundation grants have almost exclusively pedigreed people. Once you have a K nobody really care what you did during PhD. And once you have an R01 nobody cares which K pathway (if any) you used to get there.
I know people with 0 prior publications getting a K on the first submission. MANY MDs get Ks without first author papers. I know someone with multiple first author papers, one Nature one PNAS, getting his first K TRIAGED. The guy is pedigreed up his neck and savvy as hell, and I can't imagine him not shopping his grant to death before sending it in, so if his K gets triaged, it just shows you that it's basically a lottery ticket at this point.
Getting grants has a lot to do with funding priorities and who can do the work that is proposed to do. Some people are interested in doing work in a field that has little competition and a lot of money. If that's the case, funding agencies are much more willing to look the other way if you have the appropriate training and mentorship to make it happen. Other fields, such as the NCI, is a complete mess with constant single digit funding lines. So if you are 100% sure you want to do basic cancer research, you bet your ass you need the pedigree. It's just so complicated and individualized it's very hard to draw one line that works for everyone.
Overall you are right, having a 1st author paper makes it easier to establish a narrative that you have a track record of independent research. But given the chaos in the system, I don't think it's as deadly as problem as you think not having one, and I don't necessarily think that he should stay for another year. I think the idea that one 1st author paper is worth 1 year is myopic. It depends on many factors including most importantly your long term career goals. The situation is kind of complex and he needs to talk to someone with more experience to assess his situation on a more individual level.