I personally don't think it's reasonable to say that senior positions can't do a job share. I have known very senior people share jobs very successfully.
However, it wasn't technically a job share, as there had to be an overlap to hand over and sometimes there were meetings both had to attend, decisions to be made, that kind of thing. So it worked really well, partly because the individuals concerned worked well together, but it did involve an overall cost to the organisation of about 45 hours a week on that salary rather than 37.5.
The trouble is that for very obvious reasons it's more or less impossible to effectively trial a job share.
A comment on the working at home thing - you say your job is quite reactive and you get queries coming in all the time. How does that tally with you working at home, which would work best with project/written work? Is your boss absolutely right to be concerned about you not being in the office if the nature of your job means things happening there and then and having to deal with them? That sounds like a fairly reasonable concern to me.
When he said he is unable to recruit anyone else, is that because of a recruitment freeze? Recruitment does cost a significant chunk of money as I'm sure you know, in terms of advertising/agency fees, management time, loss of productivity, all that stuff.
I'm not saying he is right to turn it down, as I said I have seen it work, but it did involve compromises and extra costs on the part of the organisation. It sounds like you would be heavily reliant on a positive attitude, willingness to compromise and desire to make it work from your boss to make this happen, and it doesn't sound as though you have that.
What do you think his reaction would be if you resigned as a result? How keen is he to keep you?