There is, in fact, a filter on the end of that tube, but in my tank, the filter touches the bottom. I have seen my tank virtually empty, perhaps a cup left on the bottom, if that. If your filter is not touching the bottom, then you can adjust it so it does. You'd have to disassemble the top of the tank from the bottom half and replace/extend the tube.
So there should be no large amount stuck in the tank. There is no conspiracy by Pentair.

There's no diabolical, or practical reason, to keep a bunch of unusable acid in the tank. The feed tube has to have a filter and that filter takes up as much acid as possible. It's not going to be 100%, but it's pretty close. And as far as I know, the acid doesn't separate from the water, the 1:1 dilution remains so, throughout the liquid column, whether the tank is full or near-empty.
While I have loaded mine with four gallons, the liquid tops off in the IpH's throat, above the grate, which is above the white part of the tank. I don't think that causes any harm, but if you keep the level well below the grate, and below the seam, you only get about 3.5 gallons anyway. Pentair is being "generous" when they say it's a four-gallon tank, but technically it will hold four gallons. I suppose keeping the mixture below the seam is prudent, so really it's a 3.5 gallon tank.
Regarding the mix, whatever is left in the tank, be it a cup, or a couple inches, or half full, doesn't affect the refill mix, at all.
When I refill, I grab a gallon of 31% acid and mark its level, with a Sharpie, on the outside of the jug. I pour the acid into the IntellipH, then fill the jug with water to the Sharpie line.* Then pour that into the tank. That makes a perfect, and the recommended, 1:1 dilution. If the tank was near empty, I'll do that again, and get four gallons of 1:1. But if the tank was partially full (which it almost always is), what is in the tank is still 1:1. Adding any amount of acid, and an equal amount of water, results in 1:1. So I'm not sure what you're having trouble with, mix-wise.
I never want the tank to be fully drained, because I don't want it to run out when I'm not paying attention to it. So I throw in a couple gallons (of 1:1) whenever it fits. I've never felt that was inconvenient, since I only have to do that a few times a year. And I like that the tank only has, at most, 2 gallons of 31% in it, because if the IpH controller ever malfunctions, that's the most that could ever end up in my pool. It'd tank the pool's pH for a while, but would not result in any real damage or an unsafe body of water. I would never want to use a 15- or 30- gallon tank of acid, like some Stenner solutions. That's just asking for trouble. The "inconvenience" of a four gallon 1:1 tank is what keeps it safe.
* Pouring water into an empty acid jug, and/or pouring water into a tank of acid solution, is not the preferred procedure. You're always supposed to pour acid into water. But that's not practical for me. Trying to mix the acid into water, like in a bucket, and then pouring that into the IpH was much more prone to spillage, and making fumes, than the way I do it. I wear protective gear when I fill my tank. A full face shield and a breathing mask. It's the safest MO I've come up with. I have had splashes of acid-water come back at me while filling the tank, so I highly recommend a full-face shield.
That all said, you can avoid ALL of these issues by buying and using only 14% acid (sold at most big-box stores). That's a good dilution, and you never have to deal with mixing or diluting or whatever. Just pour in enough jugs to fill the tank and properly store the leftover (outdoors, somewhere safe).