I'd guess most are more concerned that if they give you a private office, others will request one and they aren't equipped to give lots of private offices. And if they give just you one, it will cause resentment in the team that isn't worth it.
I think that's very true, but it makes the situation even more frustrating. Going all the way back to books like Peopleware, it's been well-known for a long time that even in dense urban areas like Manhattan or San Francisco, it is more cost-effective to provide private and quiet space for knowledge workers than to attempt to "save" short-term money by buying open-plan office space and increasing worker density.
There's a ton of quantitative evidence supporting this. Yet employers persist in only ever buying open-plan space. Established companies that already had private offices sometimes even spend money to tear down the privacy features and remodel the office to function as a type of fashion signalling that they are "cool" like what they perceive start-ups to be. So they actually pay money to destroy productivity, so they pay money to move their business into a state in which the business makes less money.
Then you also see firms spending money on needlessly opulent lifestyle masturbation, like moving their office to some expensive downtown location with a roof deck. Or building a rock climbing wall in the office.
So you know it's not an issue of lacking the money for buying real estate that would support private working conditions. It's just a lot of issues about status and opulence.
So while you're correct that given their current office layout they might be unable to accommodate giving someone privacy, it's disheartening that they were ever in that situation to begin with since it would have been demonstrably more cost-effective for them to have built privacy features into the workplace from the beginning.
It has really caused me to see office space as a major red flag about a company. It says a lot about what the management of that company cares about.