Seriously, why not get a Raspberry PI 3, or something. It will be considerably more useful and accessible (I/O) + relatively future proof.
It's also considerably less open or documented. Good luck getting anything useful from Broadcom.
On the other hand, the PC has been around long enough that all the warts and quirks are well-documented, you can find datasheets for most of the critical components easily. Older ones will be better if you want to play around, especially those before the introduction of "secure" boot and the whole (U)EFI mess.
I agree that the Raspberry PI 3 is NOT fully open sourced. BUT I use it myself (version 2, currently), because I have enough information to do what I want with it, and DON'T need to know how everything about it works.
But if that worries you and/or the OP, then there are plenty of open sourced, similar microcontroller based, boards available, many of which are very reasonably priced. Or even buy a microcontroller and build it yourself.
But an old Pentium 4 system has so much wrong with it, as regards physical size, power consumption, lack of availability of brand new stuff for it (e.g. the capacitors on the motherboard will now be, how many years do you say ?).
I don't really agree with your implication that a Pentium 4 PC, is hugely open sourced and known about, in minute detail. (Fully, PUBLICLY).
I.e. If you used Dos or later (original), have Microsoft released the source code etc for it ?
Have Intel released the full factory documentation with all the (in all likelihood), hundreds of minute errata with the Pentium 4 cpu ? (to the general public).
Does it really matter if you can get the raspberry PI 3 to do what you want, that it is NOT fully open sourced ?
tl;dr
I still think that it is NOT a good idea to go for the Pentium 4's.
Allowable exceptions (in my opinion), would be a third world country, where even low cost microcontrollers, are unavailable and/or out of reach, or doing it purely for fun, obsession with intel Pentium 4's or just as a big learning exercise, would NOT sound too unreasonable.
If people are really that concerned about the open sourced nature of things, then go for a fully open sourced cpu.