Not pointed at me, but as someone with 2 MSc's I may be able to offer insight.
I enrolled into my MSc at LSE (History) with the idea that a strong BA, an MSc (Political Science) from a top Dutch university, and greatt projected grades would be enough to bag at least one PhD spot. I ended up applying to 4 US colleges while undertaking my MSc. I was naive. When I did not get in, I was disappointed but not too surprised and focused on finishing my Msc degree in History (which I thought was the missing piece).
This application cycle, I was much more prepared: two Masters under my belt, reached out to all relevant POIs, a distinction (gpa 4.0), and a prize while having worked closely with one of my LSE professors in polishing my research project ,and getting great letters of recommendation (don't want to namedrop but a top Economic Historian wrote a fantastic letter for me).
Still no chance. I will chalk this one up to COVID messing a lot of funding up, and it thus becoming a very competitive year. Nonetheless, in my experience, though unsuccessful, applying when I was done with my MSc degree, having all my grades, connections, and having the time to properly think about proposed research, felt much better.
Disclaimer: I was lucky enough to do parttime work which allowed me to pay my bills and basically work fulltime on my application for a couple of months.