Throwaway because old-timey founders are terrified of the internet.
I tick a few of your bullets, as I was hired for a local-only position which I eventually turned remote and also did all of the technical hiring (which was semi-local and 90% remote).
I applied for a "stealth" healthcare start-up "Director of Delivery" position which was local-only in a popular tech hub in the northwest US. When interviewing, I discovered I would be directing a team based in India and would be the first direct technical hire. The founders were unhappy with the outsourced work, and my job was to (1) get their abomination of a platform functional or rewrite it, (2) start hiring local talent, and eventually (3) phase out the India team. Pretty typical stuff.
Didn't talk about remoting in the interview, as it was not a priority of mine. The office was 15 minutes away from where I lived. Worked for the first few months in-office rewriting the majority of the platform, but was constantly distracted by questions and meetings. Basically became the go-to support for anything that beeped. Was operating anywhere between 30-50% efficiency in a given day. Founders came from the old school world of insurance, so everything had to be face-to-face and we had to have meetings for every single discussion. The idea of remoting was completely foreign to them, not something they were interested in talking about (I casually mentioned it once or twice to gauge their reactions).
Got to the point where the distractions were too frustrating and I asked the CEO if I could spend 2 days working from home so I could focus. He agreed. Set up JIRA and measured/tracked everything; my productivity and the India team's. After 2 months, presented the reports to the CEO. I had rewritten our platform, was ready to migrate from the old into the new, while the India team had only managed to close a handful of non-critical bugs. Explained that this was largely due to being able to work at home. Asked if I could go (almost) full-time remote with the understanding that I would be (1) available to come into the office as needed for any important meetings (e.g. C-level meetings with investors, partner companies, etc.) and (2) we would schedule a once-a-week sit down with the CEO and COO to make sure we were in sync. That's how I made it happen.
As for hiring, I got sick of working crazy hours and realized it was time to get some help. My personal opinion on remote (at that time) was that it had to be ineffective and generally a pain in the backside from both a managerial and technical/infrastructure. Despite my own experience and evidence to the contrary, I was wary of hiring someone halfway across the country. As such I opted to try and find people who were local, but wanted to work remote with the understanding that occasionally ("about once a month" is what we ended up with) they might need to drive into the office for important meetings. CEO approved it. As for how far away an employee could be located, I asked each in the interview what they would be comfortable with driving-time wise considering they had to come in once a month. Our farthest remote was ~60 minutes depending on traffic, average was ~35 (of 8 engineers total).
I answered a few of your questions in the above story, but will answer the ones I missed or was vague on bullet-by-bullet below:
* Why does the person who does this job need to be on-site? Please be specific. Give me examples of things that can only be done if she were in the office.
Because old-school management and business practices were being used by 50-60 year old founders. There was no practical reason for an engineer to be on-site, save for maintenance of one of our few on-site servers (which I out-sourced to a consulting company at some point).
* How has the policy affected your company’s ability to attract candidates?
I think our job postings attracted a very specific type of employee: the one who wanted to work remote, but wanted the ability to see and interact with their peers on occasion. We rarely received resumes from people outside of that group. Unfortunately I can't say if this increased our attraction in general, as the only engineering positions we ever interviewed for were remote.
I can say that, by word of mouth and networking (which I don't consider a reliable statistic), our remote policies definitely increased out attractiveness. Every one of my peers or acquaintances from conferences and such had great things to say about it and always asked to be notified when we had a position open. We hired about 50% of our team that way.
* What would it take for the company to change the no-telecommuters policy, even if only for one specific position?
Though I did not mention it above, our remote policy applied only to the engineering team. Our account managers, project managers, sales team, support team, etc. were in-office. I doubt they will change that; the founders simply cannot wrap their minds around how to manage a team remotely. Tools like JIRA and Slack don't make sense to them (even email is a challenge). If they did allow it, it would be case-by-case and role-based, while probably not allowing 90% remote. Example: there was very little reason for our social media person to be in-office since everything she did was online. I could see her position allowing remote most of the week, if she chose to fight for it. But the onus of proving that it would be a smart business decision would be on her; nobody from the management team will wake up one morning and think, "I should let more people work remote."