> Any entity strong enough to blackmail you into bailing it out, is strong enough to demand for the bailout to be no strings attached.
"Too big to fail" -- if they genuinely are [0] -- banks are not in a position to blackmail anyone into bailing them out; the danger of them failing is indirect effects of that failure, not actions the bank triggers if it is not bailed out -- its not blackmail. While policymakers have a strong public-interest incentive to prevent them from failing (that's literally what "too big to fail" describes), they have a similar public interest incentive to take steps to make sure that they aren't in the position to choose between similar failures and costly bailouts in the future -- and, quite often, the way that manifests is through conditions on the bailout which can extend as far as government taking control of the institution it is bailing out.
This is born out by the past history of US government bailouts of institutions that were deemed to critical to the economy to fail, many of which had management conditions attached, and some of which extended to government taking significant control of the entity for some period, and those who previously controlled the entity losing out -- the entity (itself a legal fiction created by government) was preserved, the actual people the entity belonged to were not always bailed out.
There are times when bailouts had no or weak conditions, which may be a policy failure, but clearly does not demonstrate that imposing conditions on bailouts is impossible.
[0] "Too big to fail" can also be an pretext for policy-maker favors to people with whom they have personal, social-class, or other connections, rather than a genuine fact, in which case its still not blackmail, but a whole different problem.