I am very sympathetic to your specific plight! I've never encountered Australian Huntsman spiders but I'm pretty sure I had several (dozens) of a (somewhat) smaller Huntsman spider species in a previous location where I lived. (The biggest of these were still roughly the size of my hand!)
I don't have general arachnophobia! For every other variety of spider I encountered before, I had been perfectly content to let them build their webs pretty much wherever they wanted – free pest control!
But Huntsman spiders (or whatever spiders I was seeing) don't build webs – they chase their prey, i.e. they move very quickly! They're also big (spiders).
I killed the first few – one of the first got into my kitchen sink and then couldn't climb back out. (I used a golf club to crush it.)
Several got into my bedroom and would, annoyingly/terrifyingly, stand/stick to a wall right next to my bed. A few others would get onto the ceiling (which was impressive for such big spiders), several times right over a part of the room under which I would (normally) walk. (And the ceilings in this house were pretty short.)
I did have something like a BugZooka, but it only worked for much smaller creatures, and, even then, some creatures, e.g. houseflies, and moths, seemed to be able to either fly away, or hold onto whatever surface they were on, against the suction. (It also tended to kill anything for which it did work.)
I was able to use a hand vacuum cleaner to capture the big spiders a few times, but only them smallest of them (or any that had died).
After I saw the first few big spiders, I did some cursory research, first to try to identify the spiders (which I think were a smaller-than-the-Australian Huntsman species). That reassured me that the spiders weren't a risk to me (beyond discomfort).
Then, somewhat curiously, the frequency with which I encountered them – sometimes there were several in my house at one time – forced me to to be exposed to them and exposure therapy is (AFAIK) the best treatment for phobias. (As a therapy tho, you'd typically gently increase the level of exposure over time.) Despite some very close contact, I was never bitten (AFAIK) and they were never threatening or aggressive to or towards me. My discomfort never entirely abated, but I did definitely learn to live with them to a large extent.
I also watched several videos of people, including some in Australia, 'catching' Huntsman spiders by hand and then releasing them outside. That helped me, intellectually, overcome some of my fear, tho I never tried deliberately handling them myself.
Except for their tendency to spook me by moving quickly (especially in the dark) or hanging from walls or ceilings, I appreciated the free pest control. I learned to mostly ignore them (or gently shoo them away).
But then I noticed that they all would fairly quickly die. (Several times ones that were hanging from a ceiling would die – and remain on the ceiling for a few days afterward AFAICT.)
Trapping/capturing them tho was tricky. They're fast – and big! I accidentally killed several trying to capture them, e.g. in a big plastic container. What I finally settled on was using a large plastic bag (e.g. a gallon sized food bag) and often some kind of 'paper ladder', i.e. some paper towel or a strip of toilet paper squares, so that the spider could climb up into the bag (with the encouragement of some gentling shooing on my part). This really only was possible when they'd get stuck in something like a sink or bathtub from which they couldn't climb out. Once inside the bag, I'd mostly close it, then take it outside and release the spider.
Sadly, I released several spiders outside only for them to be immediately eaten by other creatures!
For the largest spiders found in Australia, I'm not sure my capture method would work. I'd probably try to gently shoo them outside if feasible. Or, if I couldn't shoo them outside, I'd find something with a long handle with which I could either move them or crush them (and then clean it easily afterwards).