Like any good mystery, there are multiple suspects for why this error plagues Hitman: Absolution specifically:
Some antivirus (Avast, AVG, Norton) quarantines steam-api.dll as a false positive. hitman absolution steam-api.dll error
This is the Gold Standard for fixing missing DLLs on Steam. It forces the Steam client to compare your local files against the master server files and re-download anything that is missing or broken. Like any good mystery, there are multiple suspects