Yes, there are alternatives to GDP, but context is key.
If you just to make more meaningful charts or rankings of top 10 countries, you just switch to another indicator and that's it. This could also impact, to some extent, real economic choices.
Other applications are more difficult.
For example, debt substainability. Governments issue bonds to finance stuff, but they eventually have to repay at least interests. This introduces a link between GDP and debt growth. If you replace GDP with another index things get more vague: a country may have the maximum happiness, but if it issues bonds and it cannot repay its interests, it's going to have some trouble anyway.
Monetary policy is also difficult to decouple from GDP. Alternative indicators often mix many aspects, which means the central bank can either
1) target the whole index, which means it could try to influence not just interest rates, but education, healthcare and so on (it essentially blends with the government), or
2) keep targeting inflation, but through non-GDP estimates, which means the price "signal" is mixed with other "noise".