There's a study on resistance trained males with 3g of GABA before heavy compound movements - showing a large increase (but transient still means jack all for any hormone). It may be an individual thing - I tried it and did not get sleepy at all. AAMF, I've also done 5g fasted before cardio - it is definitely crossing my Blood/Brain Barrier, as within 5 minutes I get severe tingles and shortness of breath (only lasts for about 5 seconds) - still no sleepiness.
I am actually a fan of GABA - but to my understanding the tingles and shortness of breath are peripheral and do not indicate it has crossed the BBB.
I think there are two factors that people don't consider here though. If your body needs to shut off peripheral neurons using GABA, and you have a limited supply, that may mean less material inside your brain for sleep. Taking exogenous GABA and shutting those neurons off may have a sparing effect on endogenous GABA levels.
The other factor is that we often forget that we have feedback loops. Relaxing your body is a great way to send signals to your brain that it is time for sleep.
In other words, the entire BBB thing with GABA is overdone. IMO, it doesn't even need to cross the BBB to have some positive effects.