It's not your USB disks that are not recognized, it's your partition format that aren't.
So, be careful to format the USB disk (or key, or whatever, but not SD card), with a FAT32 format, nothing else.
It's a very very good thing that they removed the option "show all disks", as some users installed the live ISO directly to the disk that was in use, and obviously, broke all their installation, being unable to boot again in the OS they used to run Unetbootin.
If you installation medium is an USB card, you can use your Android phone, or an Android OS on PC (Bliss-OS/Android-x86, Phoenix OS, PrimeOS....), and run Etchdroid or DriveDroid, or other similar apk apps.
Even if it was possible, I won't trust Unetbootin to install ISO to SD Card anyway, as it was not made for this.
For instance, I'm using right now Unetbootin under another LiveCD OS.
I cannot boot my PC anymore, so I've made a Multisystem USB key, and from it, I use either Unetbootin or Multisystem-program (included in this distro) to make a bootable USB disk.
Unetbootin is for only one OS, and Multisystem can make a multiboot, with several OS, and you can add repair tool for instance, if they come as ISO (boot-repair, ISO of antiviruses, etc...), or Yumi on Windows, etc...
I used Gparted from this liveCD to partition the USB disk that I want to use as an installation medium. I partitioned it, and gave to the first partition (the one we'll use here the FAT32 format). I use the first partition, as I'm never sure that utility like Unetbootin sees the other one (I think they don't), it's the safe way.
I labelled it "FAT32" to be safe.
Whatever you did, whether you followed the first part or not if you already had a sane FAT32 partition, you have to make sure the partition you want to use is mounted.
Sometimes, you think that Unetbootin doesn't detect the partition, and it's simply that's not properly mounted. Happend to me several times.
To mount it, you have several ways, you can mount it or check via the terminal, see here for instance, it's very detailed: https://www.thegeekstuff.com/2013/01/mount-umount-examples/
I use the GUI method, as Multisystem allows me to do so very quickly and easily. As it's based directly on Ubuntu, it has the program called "Disks" (Full name: Gnome-disks-utility), the only think I have to do is to click the arrow to mount a partition, and it's done.
Usually, just plugging the USB make it mounted, but it depends on your distro, and things are not always like it is in theory, sometimes it's not automatically mounted, we don't know why.
So, when you have your FAT32 partition, mounted, don't unplug it, as it would make it unmounted, and you'll have to mount it again, and the id of the partition could also change.
I'm a little paranoid, so I always use Gparted to check visually if I'll be working on the good partition. Note the id of the partition, for instance, mine is sdc1.
I have 2 choices in Unetbootin, USB or disk in the field "type", for your type of medium, click both, and search for the partition id.
I had to close and start again Unetbootin to see my sdc1 partition in the field.
That's it. I burned an ISO to my USB external disk with Unetbootin, so whoever tells it's not possible is wrong.
The only problems that can arises are not sane partition, not rightly formatted partition, and not properly mounted partition.
After that, I make sure that my OS is able to boot directly from the USB I've just created, I launch it, and I install the OS I burned to my hard drive.
For your question about better alternatives, there are tons of alternatives, but if you have problems to use Unetbootin, you'll have problems to use them too, as they are not easier, or better, some are equivalent.
For Windows, there are tons of sofwares, but the one I use and consider trustable, it's YUMI, Multisystem or Sardu. They are the more famous, and have the advantage to allow multiboot, but you will find a lot more of them.
Unetbootin is really cool as it's simple to use, but often I prefer multibootable ones.
Even if FAT32 are the safer format to use with Unetbootin (and tools like this in general), some users use workaround: https://forums.linuxmint.com/viewtopic.php?t=165389