In Carrying the Fire, Michael Collins observes humorously that Apollo 8 was only the third time the Saturn V had flown. To a non-specialist, it does seem remarkable that NASA was confident to put humans on the mission, and to perform the first translunar injection at the same time. (I've grown up in an era where NASA seems more cautious.) Was there simply enough data from the unmanned runs to have confidence in the system? (Is translunar injection not as difficult as it seems?) Were people concerned about the pace of the program?
As I answered to another question:
Both time and budget pressure forced the program to take risks that they might not have in an ideal situation. Flying live crews on Apollos 7 and 8 was part of that risk.
Initially the plan was to fly a CSM and LM together in Earth orbit for Apollo 8. The LM was behind schedule and not ready for a test flight at the end of 1968, so rather than repeat the CSM-only LEO flight of Apollo 7, George Low had the idea of making Apollo 8 a lunar orbit mission. Almost everyone who heard Low's proposal had more or less the same initial reaction you did, but on further consideration, everyone eventually decided that there was enough confidence in the system to take the risk.
It was, certainly, risky to fly the Saturn V crewed at that point; longitudinal ("pogo") oscillation in the second stage was still a not-completely-solved problem, and lots of things could go wrong with the booster, but at every step of the way, there were contingency options and backups.
If the booster failed before reaching orbit, the launch escape system could pull the crew to safety; the LES had been demonstrated on 4 test flights.
If the S-IVB stage failed during the translunar injection burn, or if the burn somehow went wildly off-target, the main engine on the service module could be used to abort and get into a proper orbit for Earth reentry -- the Apollo CSM had a tremendous amount of maneuvering capability.
The service module's engine had already been well demonstrated on Apollo 4, 6, and 7, with eight separate burns on that last mission. If the service module's engine failed to fire to get into lunar orbit, the small RCS thrusters on the spacecraft (themselves highly redundant) could be used to make the course corrections needed to return home.
The big no-backups risk was, of course, that the service module engine would fire to enter lunar orbit, and then not fire for the trans-Earth return. Note that the engine would normally fire for a mid-course correction or two on the way to the moon as well, so the crew could have good confidence in its behavior by the time they reached the moon.
So in the end it was a combination of your guesses: there was a fair amount of confidence and data; translunar injection wasn't exactly easy, but anything that went wrong was correctable; but mostly, the pace of the program required some calculated risks to be taken.
As a complete vehicle, it was only the third flight. But the individual components had many previous flights, or weren't involved in the TLI. Let's look at the components from top to bottom.
Launch escape tower. Prior to Apollo 8, this system had 2 launch pad abort tests, 4 flight abort tests on the Little Joe II rocket, and 2 abort tests on Saturn I flights. Suffice to say, confidence in the escape tower was high. By the time of the TLI, it would have been jettisoned.
Command module. There were 5 unmanned flights of the Block I version of the CM. Many of the modifications between Block I and Block II version weren't much of a risk. Regardless, they passed qualification testing and were used on Apollo 7 without issues. This included
- a new hatch to address the Apollo I fire. This was a considerable risk, but had also been thoroughly tested.
- a docking mechanism to dock with the lunar module. This wouldn't even be used until Apollo 9.
- the outer surface was aluminized Kapton, instead of painted white.
- the covers of the scimitar antennas were proven unnecessary and were deleted.
- more connections were added to the umbilical.
- there was a slight rearrangement of the RCS engines.
- the flight computer was hardwired rather than plug-in, which actually improved reliability.
Service module. If something happened during TLI, this had the engine that would get the astronauts home. Like the CM, there were 5 unmanned Block I flights and the manned Apollo 7 Block II flight. Major differences between Block I and Block II:
- some changes in tank geometries.
- relocation of some fuel cells and their tanks.
- radiator panels were moved to different sectors.
- different paint job. However, this was the first flight with a high-gain (dish) antenna. The omnidirectional antenna had been used on previous flights and could be used as a backup.
Lunar module. A non-functional test article (LTA-B) was present on Apollo 8 to simulate its weight, center of mass, and structure. A full-fledged LM wouldn't be necessary until Apollo 9, and one had been tested in Earth orbit anyway on the unmanned Apollo 5 flight.
Saturn-Apollo adapter. This is conical part that covers the LM. Previously flown on Apollo 4 and 6, so Apollo 8 was its third mission. Not really an issue until Apollo 9 practiced docking and extraction.
Instrument unit. This controls the Saturn V stages. There were 13 launches before Apollo 8.
Saturn S-IVB third stage. This was the stage that actually did the TLI, but it had a successful history. Flew 4 unmanned missions as a Saturn IB second stage, 1 manned mission (Apollo 7) as a Saturn IB second stage, and 2 unmanned missions as a Saturn V third stage.
Saturn S-II second stage. Apollo 8 was the third flight. In an emergency, either the launch escape tower or the SM would be used for an abort. Discarded before TLI.
Saturn S-IC first stage. Apollo 8 was the third flight. Failure would result in an abort using the launch escape tower. Discarded before TLI.
So nearly everything (except the high-gain antenna) had at least one previous successful flight.