Okay. With no corals, I'd suggest that you could do slightly larger water changes to rid your system of old tap water. I'd suggest 10g - 15g every 3-5 days. You'll probably need at least 20 + changes to truly dilute or rid yourself of the tap water. Please make sure that your water change temp matches the tank so it doesn't shock your fish and keep testing to make sure you have nitrates (you.never want 0 nitrates!). You don't want to change so much that you send your system into a mini cycle - this is especially important if your system is newer. It's about finding the right balance and being patient. The process will take a while.
Out of curiosity, why did you decide on a vlamingi tang? They get quite large...
As for white spot being in the tank, if you do start to notice spots again or believe that it may be affecting your fish and you don't want to QT them again, there are a few things that may help. Firstly, healthier fish will fight it off better, so you should add vitamins to their food. I use a combo of selcon, aquaforest fish-v and aquaforest garlic oil (Use the latter sparingly). Secondly, try to maintain pristine water conditions - don't overfeed, keep your alkalinity stable and you nitrates and phosphates to a minimum. In a fish only environment, nitrates under 10 are ideal (corals are more picky). Phosphates under 0.05 will help keep algaes at bay, which helps since some algaes can really eat up the oxygen in your tank - especially free floating algaes. Thirdly, you can try feeding them medication - I combine metro and focus. Lastly, you could try something like polyp lab medic. Really, I think it only helps make things more slippery, making it harder for ich to attach to the fish (it's a water conditioner of sorts - probably helps the fish produce a bit more of a slime coat too).
Speaking of slime coats, since you do not have corals at the moment, you could probably keep your alkalinity a bit lower. The higher the alkalinity, the less slime coat your fish will produce.
Honestly, I'd consider a QT system. I get what you're saying and despite extreme effort on my part, I still managed to introduce ich into my system; however, having the QT allows me to observe incoming fish to make sure they are healthy, eating and free of disease before adding them. Since there are many different strains of ich, for instance, logic dictates that it's best to try to limit additional disease from entering our systems even if it's already present, not to mention that if the fish is eating and healthy before going into the DT, it's less likely to attract the ich already in your system. I can also pull fish out and treat them if something goes awry. The LFS QT process is likely not that great - especially if it's the same LFS that's been giving you the bad advice.
If you think about it, you can buy a 20g package (tank, heater, HOB filter, LED strip and air stone) for less than the cost of your average marine fish. I picked mine up for $80. Just food for thought
One more quick question. Do you have a sump? I note that your landscape is quite minimal, so I'm wondering what you're using to filter your aquarium??