In the late '70s and very early 80s it was not unusual to make BIOS source code available. Apple did indeed do so; the full source listing starts at page 76 of the Apple II Reference Manual. Atari did the same in their Operating System Source Listing section of their Atari 400/800 Technical Reference Notes.¹ For CP/M machines, having the BIOS source was near essential if you wanted to add new hardware to the system that could be used by CP/M. So IBM's publication of their BIOS source code in Appendix A of the Technical Reference Manual was not unusual.
Nor does having BIOS source available simplify making a clone; in fact it may make it more complex. To avoid copyright issues, your clone's BIOS must not copy anything from the published source. This means that even if you independently come up with a similar solution for similar reasons, you may have to rewrite your solution not to be so similar to what the published source does because you could be accused of copying it from the original source. Having source code with comments increases the scope of work that can't be copied, as compared to just object code.
Nor was it incompetent not to foresee, before the release of the first IBM PC, that clones would eat IBM's lunch. Nobody at that point even could be sure that a single standard system for microcomputers would ever become a thing; it certainly hadn't happened up to that point. Even if someone had had such amazing precognitive abilities that they could have seen this, I'm not sure that IBM ended up making less from their fraction of a huge PC market than they would have made from the totality of a much smaller IBM PC market that they owned exclusively. The whole point of clones was that they were significantly cheaper, and if the market couldn't move to cheaper by cloning PCs, it would have moved to cheaper by using less-compatible hardware and moving the compatibility burden to software developers.
¹Of the early PC manufacturers, Commodore appears to be one of the very few that did not distribute their BIOS (which they called "KERNAL") source, though it later leaked out. Perhaps this was because it was somewhat more substantial than other early BIOS code, being a bit closer to an actual operating system.