In my case, the presence of green hair algae (GHA) is 100% dependent on my nitrate level. Unfortunately you can't trust a test to confirm your nitrates at 0 because, as others have said, the algae could be using all available nitrates. It's come to the point where I gauge my nitrate level not by what the test says, but by how frequently I have to scrape my glass. The amount of GHA is a more accurate test for nitrates than the test itself, lol.
Personally, I achieved truly low nitrates and the elimination of GHA by adding a sulphur denitrator. It is the lowest-maintenance way of keeping nitrates at zero that I'm aware of, and is the most important piece of equipment I have related to the elimination of GHA. Some people will tell you that you'll never eliminate your algae issue without pulling all that stuff out, maybe even scrubbing your rock, but in my case the large mass of GHA simply melted away within a few weeks of dialing in the denitrator. I think my protein skimmer grabbed a lot as it was dying off.
The next most important piece of equipment to keep the algae out in my case is a UV sterilizer. For your size tank, however, you'll need a significant amount of wattage to make a difference, which is maybe why you haven't had much success with yours. It seems like most popular devices are in the 5W to 9W range, which are simply not powerful enough to do much good in saltwater. 9W works good for me, but my tank is about a third the size of yours.
The next most important thing related to algae elimination has been the addition of a diamond goby. Any sand-sifting species should achieve similar results, but diamond gobies run a little larger than most and are therefore more efficient at keeping the sand clean. In my experience they are far superior to crabs or snails for cleaning sand, but snails are still necessary for the glass. I used to have brownish sand like yours all the time, but disappeared completely within 2 days of adding the goby, never to return. Then the goby died (in hindsight, I believe I was underfeeding him) and the brown stuff on my sand started coming back within two days. So yeah, I'm a big fan of sand-sifting fish.
Having said all that, the main reason I (we?) use all these gizmos is to cut down on the amount of water changes we do. The fact remains, however, that water changes are the most effective way of eliminating most algae issues. What would probably be much more effective than buying yet even more equipment and stressing your wife out in the process would simply to be better about water changes. I know it's the advice that nobody wants to hear, but it's there for a reason. And, as you appear to have learned already, NEVER use tap water for those water changes.
It's not a hobby for everybody. I wouldn't be able to stick with it if I didn't get a significant amount of joy from the challenge, as opposed to only getting joy from gazing at a fully stocked, beautiful algae-free tank. My wife too put pressure on me during those times when it was an embarrassment visually, but she eventually had to get used to the idea that this was a passion for me. Fortunately for both of us, that passion eventually translated into a tank that solved it's algae issues and is now a joy to behold (although I still manage to kill things from time to time). If the challenge doesn't bring you joy...Good luck!
