I feel you might be mischaracterizing the argument.
As I've recently discovered (I lucked into another game's Beta), nitpicking of wording is important in RPG books. Grammatically and through common sense (including context clues), I could approach the "bonus", "career", and "specialization" skill argument from a couple different directions, for and against. (I'm happy to share if anyone cares.) That I can see different arguments means there is room for debate, and no one here is being "silly".
I would encourage fellow GMs to resolve it as benefits their games. My current campaign features a small party, and I would be loathe to limit their ability to broaden their skillsets. However, if I were to run another game that has more players, then I would strongly consider limiting them to only gaining the "Specialization Skills" from any new Specializations they purchase.
I agree with all of this. I'm in no position to say I'm right, or wrong. I just assume that when one purchases a specialization within their career, that all they really get is the bonus skills, that it would carry to the non-career specialization as well. The extra xp spent on specialization is for the ability to go down a new talent tree, as well as the bonus skills that became "new" career skills, thus saving you 5xp a purchase further down the line.
The xp saved by only giving the bonus skills would be, 4 skills at 5xp per rank, so 100 xp saved. By getting all career skills and "bonus" skills, that's 12 skills, a saving of 300xp. That's an impressive savings for the ten xp spent on the non-career specialization. This also assumes none of the skills are in the the first career, or first specialization.
If my math is wrong above, feel free to let me know. I'm, by no means, suggesting "I'm right", it's just how I see things, and let's be honest...I've played only long enough to buy one extra specialization...