While we don't have an exact handle on the risks posed by the vaccine long term, we do have reasonable bounds on what they may be. For example, it's entirely possible 0.05 - 0.6% of people who receive the vaccines develop long term severe negative outcomes. Perhaps that number is 3% - we don't know, but more unlikely. We can be fairly certain that number isn't as high as 40%. We don't have all the numbers, but we have reasonable assumptions - these are non integrative mRNAs that have a very short lifespan inside the body, so long term effects other than immunity are not expected. Of course, there is uncertainty in everything.
This is more uncertainty than the FDA would accept in the usual course of events - but this is an unusual time. The FDA has no adequate, approved, and available alternatives to the emergency authorization when faced with the pandemic. The investigational vaccine appears to be safe and effective against COVID-19 for the general population. Yes, the vaccine poses some unknown risks, but they are very likely vanishingly smaller than the very present health risks posed by the pandemic. For any given individual, they're much more likely to suffer negative health outcomes within the next 5 years from COVID than from the vaccine. Nobody is stating that risk to be zero. But that also doesn't mean "the risk is completely unknown so why even bother?"
How is the emergency authorization any different from an approval? If tomorrow an effective treatment for COVID-19 were to be approved or the pandemic ends, the EUA for the vaccine would be rescinded. Pfizer/Moderna must also track any adverse reactions more closely, and report them to the FDA on an ongoing basis. They also cannot sell them to private players - they're being provided to the FDA & their authorized parties. The vaccines are not licensed for COVID or any other use.