Second Impressions? Following up after a bad PE fit interview

Context: I'm in lateral processes for a VP role. One process was with a PE team at a bulge bracket organization (GS, JPM, MS, etc.) that was concluding after having gone on for 4-5 months. It felt like a great fit with the team, but as a result of some personal matters (family health issues), I had a tightening timeline for relocation that I was working with.

The team itself was really responsive to my timeline, but HR at this organization was not. Long story short, the internal recruiter ended up calling me to schedule the same day fit interviews with the HR team covering the team that was hiring. I'll be the first to admit that I was not on my A-game as my patience with the team after so many months had worn thin and I had to switch from dealing with family matters to jump into another full set of fit interviews on short notice. I was later told by the team that while they wanted to hire me, HR shut down my ability to move forward in the process.

I don't think what they saw in those interviews were representative of who I am / the cultural fit I'd be (especially after so many months of investment in the process and the team's response), but I also recognize the value of just moving on from a rejection. Has anyone else dealt with something similar? Are follow-up emails owning up to and contextualizing my mistakes/flubs helpful in this case to send to the HR team if I want to keep the door open with the BB down the line?