With the traditional 5 stage tank RO system the faucet usually slows the water down to a trickle.

Does this have the same problem? I want to fill up a 5 gallon water jug to use with a water cooler so I'll be running it for a while.

Since it's tankless it won't slow to a trickle but it'll take a little bit of time to fill up a 5 gallon water jug.