That... seems high?
I couldn't find any more info with a quick search as the results were polluted with articles about measles deaths, which also mentioned vaccination. Do you have any links to hand with decent stats for that, or a good set of search terms I could use to find out more myself?
According to the stats [2], there were 134 million MMR vaccines administered between 2006 and 2022, and 28 "compensable concession" claims -- where the Department of Health and Human Services determined that compensation was warranted; and an additional 124 claims where they ended up paying but don't necessarily admit that it was warranted (either ordered by a court or settled out of court). That's about 1 in 1 million for claims paid out, and 1 in 4 million for "concession" claims.
There's no indication in [2] of exactly what the claims were for; looking at [1] it could be anything from death from anaphylactic shock to even include things like "Shoulder Injury Related to Vaccine Administration".
This page has some investigation about deaths related to various vaccines, including the MMR vaccines [3]. A lot of the deaths were related to children with a compromised immune system or unsanitary conditions; the report says that there was evidence that MMR sometimes causes anaphylaxis, and that anaphylaxis can be fatal, but that they didn't have any actual instances of MMR causing fatal anaphylaxis.
Regarding the original point though: even if deaths are due to unsanitary conditions or compromised immune systems, and even if modern medicine can often successfully intervene to prevent death in the case of anaphylaxis, the fact is that injecting things into the body has an inherent risk, no matter how small. But the risk of not doing the injection is far higher. We'd like there to be an "always safe" option, but that's not the world we live in.
[1] https://www.hrsa.gov/sites/default/files/hrsa/vicp/vaccine-i... page 2
[2] https://www.hrsa.gov/sites/default/files/hrsa/vicp/vicp-stat... page 3
Of course you do, at least you do if you're the head of a public institution (who should be treating lives equally). But that's probably what GP meant by "bound to kill people".
The trolley problem is just a fun/meme way to visualize the choices influential people have to make daily. [And, FWIW, like the trolley problem, the choices get harder if you have a non-"neutral" agenda, i.e. when you can't just tally the number of lives impacted and abandon the group with less lives involved.]