1) Exclude deaths that'd hurt your life expectancy. For instance, by under counting infant mortality. E.g. does a child who dies during birth count against your life expectancy? Arguments can go both ways, but if your hospital's funding depends on having good life expectancy numbers in your community... well, there you go. IIRC this is one factor that helps explain Cuba's surprisingly high life expectancy numbers, though it doesn't account for most of it.
2) Overtreat end of life issues. Instead of focusing on quality of life and general health throughout the lifespan, put all your funding into heroic interventions that let 90 year olds live on artificial respiration for months. This is what the United States does.
3) Systematically deceive. Have a significant homeless population? Rural population that's deeply unhealthy? Make sure you put a ton of resources into measuring outcomes for your urban population with relatively good health access, and the peasants you miss are just a sign of not getting enough grant money from the WHO. Governments that put resources into getting measurements of the most marginal populations are penalized, while those that deprioritize studies of those marginal populations would have better numbers for their marketing brochures. Even though we'd expect the former to have actual better outcomes than the latter!
Better to measure quality-adjusted life-years, in a way that accounts for the fact that it's sometimes better to take a risk of death with a possibility of full recovery over a guarantee of lost function.
I can't possibly enumerate all possible metrics and their effects, but I bet we could do pretty well at imagining faults for anything that can be constructed.
It turns out african americans skew our metrics because they have
1) high infant mortality 2) high murder rates of young men
The high infant mortality is irrespective of wealth and has a disproportionate impact on average life expectancy because the death of a 0 age person has a huge impact on the average. When you look at the rest of the population, our whites are on par with europe and our asians are on par with asian countries.
One way this is gamed is that some countries have different standards for reporting infant mortality.
<<The infant mortality rate is defined as the number of deaths of children under one year of age, expressed per 1 000 live births. Some of the international variation in infant mortality rates is due to variations among countries in registering practices for premature infants. The United States and Canada are two countries which register a much higher proportion of babies weighing less than 500g, with low odds of survival, resulting in higher reported infant mortality. In Europe, several countries apply a minimum gestational age of 22 weeks (or a birth weight threshold of 500g) for babies to be registered as live births. This indicator is measured in terms of deaths per 1 000 live births.>>
In the US it is gamed in reverse in the sense that the average hides the true problem is with subpopulations and not the entire US medical system as a whole. Our (gamed) poor performance in this statistic is used as a measure to support moving to a substantially different system.
Your point about infant mortality statistics being gamed is on point, though (and I pointed it out in a sibling commment).
Proxies are not intrinsically bad, either, but they have to be difficult to achieve without also achieving what you really want - when this is not the case, making the proxy measurement a target motivates people to seek workarounds.
Life expectancy might be a proxy, but it is a pretty good one.