Why maintain a false facade?
People like dealing with people. In fact, they often prefer it over dealing with a company. There's no reason to pretend you are something you are not.
I contemplated this same thing briefly when starting out. What I found was ...
As an "organization" when I received a cold call from someone, they were merely shopping around and trying to get general pricing or guidelines. Most never wanted to proceed past an initial exploratory conversation. A few did, but they were simple one-off, no relationship, jobs. I much prefer repeat business. I had to "sell" and convince these cold calls their project could be completed. If work was gained, the clients were far more rude and demanding. They failed to treat anything as if they were dealing with people and every conversation was along the lines of ordering lunch or some other non-personal conversation. It was very difficult to pick up fluctuations in perceptions and desires because everything was so cold and distant to a degree. Issues which arose were first seen as my "company" issues rather than an honest, open-minded look to se where a problem may lie. In short, as an organization you are treated like a building -- no personal connection.
As a person, people speak with far more respect and value my time a great deal more. Cold calls were directly seeking to hire me and only wanted reassurance. I had to talk myself out of the work with these calls. Clients understand that it's my knowledge and experience at work not some nameless intern working on their project. They understand that if I present an issue they can't assume it's my fault. In fact, most assume the opposite knowing I wouldn't bring it up unless I needed to. I'm given far more consideration when I need more time on a project. I'm paid much faster and with less issues than I was as a non-personal company.
In general, people treat people like they want to be treated. People treat businesses like the business is out to get one over on them. There's a world of difference.
That being posted, if you present yourself as an "organization" do you really want to speak about yourself as a third person? What if you have a distinct personality conflict with a client mid-project and they want to work with someone else? Do you then disclose your'e a one-person operation? That will result in feelings of betrayal and a loss of trust - goodbye client forever.
The only feasible reason I can see to portraying yourself as a company is to have multiple personalities and use multiple email addresses to respond to clients. What about phone calls? What if a client asks, "Why do we always deal with you? Is there someone else we can talk to?" Sure, as an organization you may be looked at for some larger projects by bigger clients. But if you can handle the project as an individual, you'll still be contact by those clients (provided your marketing is decent).
You can be a one-person company operating under a company name but its only employee/owner/operator. That doesn't really seem to deter anyone who's ever contacted me. I brand my company. And I don't hide the fact that my company consists of only me. I didn't want to brand my name (it's too difficult to pronounce for some). It's okay to be a person with an accountant, or a person with an assistant, but in my view don't pretend to be something you're not. It is way more hassle than it's worth in the long run and there are practically no benefits to it.
I will add that if you choose to go with a company branding, your name or some other unique identifier in the company name can be very helpful. For example, people may forget that "Blue Goat Coding" did that work for them 5 years ago. Bu they will remember they dealt with "Bob Smith" on that project 5 years ago. If the company name is "Smith Coding" or "Code by Bob" it will assist in client retention in the long run.