It is hard not to have to use "specialist tools" when doing special cuts. I've used sharpened screwdrivers and allen wrenches in various turning situations but definitely not to "square and smooth" small boxes.
First bore a clearance/depth hole. Cut from the center out with the flute facing 10:00. It's sort of like scooping ice cream for a cone. Take light cuts; you are not trying to pry the box out of the chuck.
If you are turning a box with a rounded bottom, use a 3/4" half round scraper with side clearance on the left to shear scrape smooth. Pull and pivot the scraper to the left with a very light cut. The scrape will be tilted to the left at about 45degrees. Only fluff should be coming off the scraper.
If you are turning flat bottomed, square inside cornered boxes and don't have a 3/8" or 1/2" scraper, try a 1/4" cabinet chisel with very light scraping cuts to square the inside corner. Make sure that the scraper is flat on the tool rest when you engage the bottom or you will have a nasty catch when the chisel slaps down.
With lighter cuts and a little slower speeds, you should be able to turn nice boxes without buying new, expensive, single purpose tools.
Most screw drivers are surface hardened only and will make a poor cutting tool, but since all steel is harder than wood it will kind of work. You will however have to almost constantly resharpen.
If I do have to scrape, just a 1" square scraper works fine (assuming of course that your box is at least that big...). I have ground a bit of relief on the left side so the back doesn't rub on small diameters. You can also do this with the long point of a small skew...
If you struggle with something not meant for the purpose it's amazing how much better standard tools can seem when you return to them. If the boxes are small, say 2" dia 2 to 3" tall at most you should be able to get by. I use a spindle gouge, sometimes an old bowl gouge at least I think it's a bowl gouge it's got a flute length of about 1" left and the grind is anybody's guess, but I can just get in there with it and get results. It's as soft as anything so I'm always sharpening it and that inch is soon going to go. I won't be able to make boxes then and I'll have to go back to the cold chisel and start over. But not a screwdriver. I haven't been tempted to try it but I feel it would be even softer and likely to vibrate like the clappers.
If you are working with the harder denser woods, Crown Tools is coming out with a new tool called a Box Tool that does an excellent job on harder woods. I've had one for about two years made by the fellow that introduced the Skewchigouge(?), sorry I don't remember his name offhand, which Crown is now producing also. The box tool is a fairly simple tool, which consists of a grind on the top about 1/3 of the way through the stock. Then a grind on the end that has about a 60 degree bevel and the cutting edge tapers to the right about 1 or 2 degrees. It is an excellent tool. Although you could make one from some round tool steel, I don't think that it will be too expensive from Crown and it is ready to go.
Incidentally, I've ground tools from old screwdrivers. The steel is poor enough that it is hardly worth spending the time at it. Unless it is a one time use and special shape that you need, then almost anything will work.
Cold chisels and cheap nail sets or pin punches might also make nice tools, if you can find them of sufficient length and if you quench them often while grinding.
I'm not saying you should make all your tools, but scraping tools are easy and other special-purpose tools can give one a lot of self- satisfaction in use.
I advise you not to spend lots of money on sharpening jigs, not to go chasing all the "this'll fix it" tools and even to give cold chisels a try for your box scrapers.
The best advice I can give is that you're in need of the only "tool" you haven't tried yet...time. There are going to be lots of bits of wood ending up as scrap before you get the facility with the tools that enables you to tackle a variety of jobs. Things go wrong all the time, if it's any consolation I have horror days on the lathe when for one reason or other nothing goes right. I don't suppose I'm alone.
Yesterday I was making some stoppers for bottles and without thinking I just kept turning the blank piece of wood to a cylinder between centres until it reached too small a diameter to grip in my chuck. Not having enough sense to grab another piece I set about rescuing the situation by trying to cobble together an alternative way of holding it. I ended up wasting time, getting tetchy and not making stoppers.
Now a four jaw chuck or another set of jaws would get me out of that problem but I'm not rushing out to get them, I'll try to have my mind on the job next time.
To go back to your point about turners being told to buy this or that tool, part of the reason is that a lot of folks on this newsgroup keep asking what should they be getting. Some have the cheapest lathes and yet ask advice about the most expensive chucks, others see or hear of the latest tool for such and such a job and think they have to have it. Lets face it some people won't be happy until they've got one of everything. Nothing wrong with that if that's something you can afford to do.
The trick is not to get disconsolate about what others may have, or be able to do, but instead keep on trying with what you've got.
It'll get better if you want it to get better and if you can recognise that it needs to get better. Some people plateau with their turning ability and are either quite happy or quite unaware at that. Others keep asking questions, of themselves and others and learn to overcome their problems but it all takes ..........time.
There is a place for making your own tools and accessories. I've made several. Some tools are just not practical for everyone to make, as an example some box making tools, scroll chucks, and the lathe itself. Some people could and do make these things. Everything is a compromise and we have to find that personal balance. Some of us have more money than time and some have more time than money and both of these groups look to each other and wish they had more of that which they have less of, time or money.
I have never found screwdriver steel to be of correct alloy to hold an edge or even stay straight if ever a catch occurs. One of my favorite homemade tools was made using a 3/8" HSS lathe bit a metal working friend gave me. I epoxy glued the bit into a 2' length of 1/2" EMT steel conduit, then poured molten lead into the conduit. I had all supplies on hand so it didn't cost me anything. I use it as is without a handle though one could easily be added. This tool is rock solid and cuts through endgrain like butter.
Sorry to say the best thing I've got for flat bottoms is the termite. No sanding required!