I think an important category has to be: no outside servers designed to help with this speedrunning challenge. Without such a rule, one can get an outside computer to do much of the work. One would merely write a small program that connects to the known IP and pulls bootstrap code straight into memory. (Granted, still a fair piece of work, but without such a rule the finish line for the speedrun would become "establish a TCP connection").
EDIT: I wonder if including the Linux kernel + C library is just too much. How minimal could one go with this challenge, yet have it still be fun/doable in a reasonable duration? You start with just MS-DOS on the disk? Or a Forth interpreter? Or maybe you start with a blank disk, but you get to twiddle bits one at a time before you first use it, in the vein of the Altair 8800?
EDIT 2: An even more entertaining idea than some sort of 8800-style toggle switch interface: you start with an actual Altair 8800, then get a few "stepping stone" computers that have just enough hardware compatibility that you can transfer data from one to the next. The final computer is a modern PC with a network connection.
Generally, speedrunning communities (for games and other things) already bar tricks and strats that require setup on an external file/system.
For instance, you can't carry hearts/stamina from one file to another in a Breath of the Wild speedrun, or use wrong warping to instantly go to a location saved on an alternate save file.
There are exceptions in certain scenes (in speedruns for Banjo-Kazooie, a glitch is used on another save file to fix the RNG for a quiz section later in the game), but generally, everything has to be done from an empty file/system with no external resources already setup.
There are different categories for everything. There's getting to the end of the game and defeating the end boss ASAP. For games with a completion percentage, there's speedruns to get to 100% completion rather than just getting to the end. There's categories for not using "warps", and others that allow using "warps" to skip levels.
Tool Assisted Speedruns (TAS) is a category unto itself. This involves running game on an emulator and making human-impossible precise set of inputs, often disassembling the ROM to find bugs to take advantage of. The Arbitrary Code Execution 'stunt' category includes one where someone programed Super Mario World by picking up items in a precise order (and time) to specify bytes, managing to write tetris and jmp to it.
Competitions are made up, so we can make up whatever point A and point B is, and the rules along the way. Timing going from a Ubuntu ISO to desktop browser with google.com loaded is one A, B pair, but there's so many different possibilities out there!
Would this rule also invalidate the OP’s approach of getting to the point where you can download wget, then using that to download other programs? wget, and the things you’d download with it, would live on external systems.
EDIT: I wonder if including the Linux kernel + C library is just too much. How minimal could one go with this challenge, yet have it still be fun/doable in a reasonable duration? You start with just MS-DOS on the disk? Or a Forth interpreter? Or maybe you start with a blank disk, but you get to twiddle bits one at a time before you first use it, in the vein of the Altair 8800?
EDIT 2: An even more entertaining idea than some sort of 8800-style toggle switch interface: you start with an actual Altair 8800, then get a few "stepping stone" computers that have just enough hardware compatibility that you can transfer data from one to the next. The final computer is a modern PC with a network connection.