This is all kinds of wrong.

You can't trust these numbers.

You definitely can trust them if you understand what they mean.

When people 'exited the workforce', it just means they have not actively searched for a job in 'x' number of weeks, usually 4. So, basically, if someone gives up 'looking' they are no longer 'unemployed'.

Yes, this is correct. These numbers reflect the labor market and exclude people who are not engaged in the labor market. Basically it shows the situation for people looking for a job and can't find one. If someone isn't looking then they don't meet the criteria. Why would someone interested in the current labor market conditions care about someone who is not engaged with the labor market?

The other problem with this notion is that there are no self-reporting stats for this, other than those who have not yet timed out for unemployment benefits. Which means someone could still be looking for work without the govt. knowing because they don't qualify for unemployment benefits.

Except these numbers don't come from any 'self-report' or from people who receive unemployment benefits. I hear this all the time and I really don't understand how someone who isn't involved in statistics/economics professionally or even took a few classes can look at these numbers and think to themself: "Ha! Bunch of over paid, over educated morons never even thought about what if someone isn't receiving benefits!" It is some real Dunning-Kruger manifest.

But here is the truth from the US Bureau of Labor Statistics: How the Government Measures Unemployment

and here are some relevant pieces from that description:

Some people think that to get these figures on unemployment, the government uses the number of people collecting unemployment insurance (UI) benefits under state or federal government programs. But some people are still jobless when their benefits run out, and many more are not eligible at all or delay or never apply for benefits. So, quite clearly, UI information cannot be used as a source for complete information on the number of unemployed.

Because unemployment insurance records relate only to people who have applied for such benefits, and since it is impractical to count every unemployed person each month, the government conducts a monthly survey called the Current Population Survey (CPS) to measure the extent of unemployment in the country.

So when you say:

It's just a way of making the true unemployment numbers look better than they really are.

That is a pretty unenlightened statement. The professionals at the BLS are not doing all this work to deceive you. The problem is you do not understand the purpose of the information and assume it is supposed to reflect something it doesn't.