Whether or not it happens seems to depend on the team. Personally, I like teaching people new stuff so it would often fall to me, on the teams I worked in. ;)
Of course, that founder was one of the originators of XP practice, and ran our company on those principles--pairing, fast feedback loops, and mentorship for all (he's since handed over the company to his #2 who he mentored for a decade, and who is continuing those practices).
The most impactful mentors I've had personally were senior developer / tech lead types. My managers were great, but it's just easier to have the "is this a stupid idea" conversation with somebody who doesn't sign your paycheck
Once I joined I learned that 30% of my time would be dedicated to hiring. The company had a pooled hiring process, and I came to hate it. I spent several hours a week reading resumes, interviewing, reading interviewers' feedback, sitting in roundtables, strategizing with recruiters, and if all that went well, pitching candidates to the hiring committee. Then I would sit back, with no control over the comp, and watch the candidate choose another team, largely because my team wasn't a "sexy" team; very critical to business, but not sexy.
In the handful of cases where my team was the only option, the candidates would turn down the offer due to "insufficient compensation". The company had a "no negotiation" stance on offers. So once the offer went out, that was that. In my first performance review my manager put "hiring" as an area of improvement. I felt this was a trap and would be used to let me go eventually, in spite of the fact my team grew from 3 to 10 engineers from internal transfers and we were hitting all our deliverables.
Between hiring and navigating the terribly siloed corporate structure to build cross-functional teams I had almost no time outside of 1:1s to spend with my own team. It was a really disappointing experience and I felt really bad when I left, because my team and stakeholders were all delightful and capable people. Like you, I felt I couldn't protect them and keep my mental health intact at the same time.
Near the end, I could only dedicate about 5% of my time to dev and < 30% to support my largely junior team. The rest of it was wasted on bureaucratic processes that were a vampire of my time. In the end, I didn't think I could honestly protect the team through what was surely to be a painful slide into meteocracy, so opted out. I felt terrible leaving them to whatever came next, but once you realize you're unable to make positive change, it's time to go.