So the basic way a Reverse Osmosis filter works is that the water is pressed into the filter by incoming pressure, some clean water makes it through and part is sent to your drain as waste water. You have to have both flowing otherwise it is not really working right. It looks like your system has a few sediment possibly carbon filter and 2 reverse osmosis filter cartridges. This allows you to produce more water than some systems. But if you want to be able to have water on demand you really need the expansion tank. Also many people have a separate spigot for the RO water, because using this to clean does no good and only reduces the flow rate. You do not want to run hot water through this filter system, so if you have your faucet on a medium temperature the way you have it setup now you will have a trickle of cold clean water and a bunch of non filtered hot water. The reason for the dirty looking water at first is simply that you need to run the system clear for 5-10 min before the sediment from shipping will be flushed out.
Hopefully once the system is all dialed in you will be happy with it. I have something like yours (a few more filters), it takes 10-15 min to fill a 5 gal bucket but my water is just water. The HD does have expansion tanks, more plumbing fun but it will resolve some of the issues you are having. Adding a booster pump will not help because it takes time to filter the water, forcing it through will create more leaks and reduce the life of the filter.
Ex Realtor, house flipper, remodelor and now building a vintage truck, making friends and fixing my house.
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