It's hard to test systematically for early instances of "they" as third-person singular pronoun because such instances often follows on a contingent phrase that is highly specific and variable from one instance to the next. For example:
If a person [performs action X], they will [get result Y].
However, I decided to check Google Books results for several phrases that are easier to test for because they appear as single blocks of words:
a person knows they
a person feels they
a person sees they
a person understands they
Expressions of this form are fairly common in recent published writing. For example, from Carol Noyes, Coming Full Circle: One Woman’s Journey Through Spiritual Crisis (2014):
I also remembered that several years earlier my chiropractor had told me that a person knows they are a shaman if they are nearly struck by lightning.
Matches for these four phrases are rare in Google Books search results from before the late 1960s, but they do occur, primarily in extemporaneous remarks recorded at administrative hearings, for example. Here are the earliest examples that involve (or may involve) use of they as a third-person singular pronoun.
From a letter dated July 2, 1802, from Thomas Jefferson to Mary Jefferson Eppes, reprinted in B.L. Raynor, Sketches of the Life, Writings, and Opinions of Thomas Jefferson (1832):
I have not heard yet of the disease [measles] having got to Monticello, but the intercourse with Edgehill being hourly, it can not have failed to have gone there immediately ; and as there are no young children there but Bet's and Sally's, and the disease is communicable before a person knows they have it, I have no doubt those children have passed through it.
Here Jefferson may mean "before a responsible adult knows that the children have it"—but to a person accustomed to frequent use of singular "they," it also sounds right in the sense of "before a person knows that he or she has it." Most of the remaining examples in my answer are far less ambiguous than this one, however.
From remarks by John McDougall on January 19, 1906, in "Minutes of Evidence Taken Before the Royal Commission on the Care and Control of the Feeble-Minded, Etc." in Parliamentary Papers, House of Commons and Command, volume 36 (1908):
The members of the committee are seeing the patients almost constantly. Persons frequently ask to come before the committee, and we see them ; the more insane they are, the more they are seen, but it is a very great comfort to them to come and be seen, and to meet the members of the committee of the committee. The patients look forward with eagerness to the visit of the Lunacy Commissioners. Everybody desires to get away from the asylum ; it is very natural. The moment a person feels they are comfortable in an asylum and would not like to go, they are the very persons to be picked out and discharged.
Here the speaker shifts from "a person ... they" to "they are the ... persons," indicating that "a person" may have given way to "persons" in the speaker's mind by the time the first "they" showed up; nevertheless as a matter of grammar, the first "they" refers to "a person."
From Clyde Kluckhohn & Mischa Titiev, Old Oraibi: A Study of the Hopi Indians of Third Mesa (1944):
These [characteristics of certain plants] were very secret things the old people knew. It is very dangerous to have all six of them. They didn't tell the persons who had these plants what would happen if you had all six, just told them it as very dangerous.
In case a person knows, they would use some of these four on you, and your mind would be all kind of losing your mind, you would get weak in mind and suffer that way gradually. The last two will put sores on you or inside you if used against you. With the first four you will lose your mind and get weaker.
I cn't tell exactly what's going on here. "they" might refer to "a person" or it might reach back to the previous paragraph and refer to "old people." Because the writing seems to be a transcription of speech by someone whose English is somewhat pidginy, the intended meaning is a bit opaque.
From Hearings Before the [House] Subcommittee on Public Buildings and Grounds of the Committee on Public Works (1947) [combined snippets]:
Mr. ANDERSON. This would nullify the experiences that you are drawing from in England—where they do not have the right to refuse [to take a breathalyzer test] if the officer stops them.
Mr. TOFANY. Except that the psychology of it is there, and that if a person knows they are subjected to that preliminary kind of screening and if they refuse anyone, they may be guilty of a traffic infraction—and if they are not in a condition that would lead the officer to the opinion that he would have to have one in order to make the arrest, I think the majority of the motorists would be reluctant to drive and drink because of the fear of that screening test.
This instance unquestionably uses "they" to refer to "a person"—although it is true that this hypothetical "person" is being treated as representative of all motorists who might find themselves in the situation that the speaker is describing.
From an unidentified article in International Stereotypers' and Electrotypers' Union Journal, volume 60 (1965):
This service is rendered to those that want to help themselves, so if a person feels they don't want to travel or maybe move to where the work is then don't bother to write. And if a person wants to be called then they must except phone charges.
Here again, we have a hypothetical person standing in for a class of users but being referred to with singular "they."
Conclusion
As noted above, one striking thing about these early instances of singular "they" is that they involve "a person" who represents a hypothetical class of people; that is, they involve a singular "they" with no known or specific identity who stands in for a plural "they"—members of a class of real or hypothetical people who share some key characteristic.
The example that the OP asks about is quite different. Here the "they" refers specifically to the person or persons on stage and answering questions posed by the game show contestants. The host presumably refers to this person or persons as "they" because he knows that they are not just a single person. The panelist's somewhat slow-on-the-uptake remark that "I think there's more than one person there" is thus a vindication of the general understanding in 1954 that "they" generally referred to plural entities.
It would not have been unprecedented if the show's host had used "they" in a singular sense, but it would have been extremely unusual, given the specific nature of the referent for "they" in this instance. In my view, that isn't what happened here, but it certainly did in the 1908, 1947, and 1965 examples quoted above.
