Immunity (Impunity?) paragraphs are themselves law. Therefore, when they apply to somebody, that doesn't constitute "being above the law" for that person.
Immunity paragraphs are the logical byproduct of the idea itself, the idea that we must govern our lives by a system of laws.
The laws in the "modern world" stem from the social contract, the idea that individuals give up the right for personal violence to a state which in turn pledges to treat them fairly, so that they actually don't need personal violence because they should, in fact, have a fair life.
Since there is no such thing as "state" with any material power and thus capability to do what it's supposed to according to the contract, it has to subcontract that part to actual people (government, judges, police, civil servants etc.).
The problem starts when you have an actual person acting in some way. If it's a "normal" person, not holding some official position, then it's clear, the person acted themselves, for and by themselves.
If it's a person with an office, then the question arises whether the person acted as an official or as the normal person he/she also still is.
For the "official" role, different laws apply than for the "private person" role. Examples are many - soldiers that may/should/must kill, police taking somebody in custody or even shooting etc. . Did the soldier kill as a soldier and earn the proverbial medal or as a private person and earn a jail sentence?
This is where Immunity sets in - since officials are likely to be watched more closely than random individuals and potentially met with envy, the immunity seems like a logical way to protect the system from a flood of suits/allegations and make it even possible for a person to step up to an official job. If a soldier had an investigation and lawsuit coming for every single killed enemy, war wouldn't make any sense. (Wait a minute...)
Holding officials to even higher standards (implied by call for harsher punishment in your question) would either require specific laws to be lawful or would itself be outside/above the law. The question arises why an individual should be held to higher standards because he/she held an office in the first place. To serve as idol or "good example"? So they can't be blackmailed? Because that's the only way we can "trust them"?
As official, specific laws apply anyway - no point in comparing higher standards to "normal people" therefore. As private person happening to hold an office, the standards are the same than for normal people, because there are no other. Like not stealing.
What could meaningful "higher standards" be? Not to cheat, for example. As an official, it seems counter-intuitive that cheating is even possible. An official position is more like an automatism, cheating is not really an option. So, a person with an official position can only cheat as a person. Now - is cheating illegal for people? Not according to "modern society". Only specific "immoral" acts are outlawed and even there prosecution is very difficult.
Due to the apparent paradox of this dichotomy (authorities to be above the law via impunity or not above the law as normal citizens) causing unresolvable problems thus rendering the social contract impossible, Hobbes chose the Leviathan - an absolute sovereign to cut through the chase. Leviathan was of course above the law.