Here is the first thing to note, video game history is a history of corporations. As a result a lot of information is lost due to internal secrecy. For instance, it was not until a few years ago that we actually uncovered the long rumoured copies of E.T. buried in the Nevada desert. The existence of this dumping was never confirmed by Atari, but we now have the evidence to back up these claims.

While I don't have time to form a more extensive answer detailing the whole games world at the time, I can answer your question more directly. Nintendo was on part of the initial push for CD based consoles. After Sony and Phillips released the CD-I in 1991, Nintendo and Sony partnered for the Playstation. This console would play both SNES cartridges and CD-roms. This Playstation was developed to the point that it was shown at the 1991 Chicago CES. However relations between the companies soon soured, and Sony released the Playstation independently with no SNES compatibility. When it came time to develop its next console, Nintendo was in a market dominated by the CD based Playstation. However, we cannot look at the CD at this point and say its dominance in the future is a foregone conclusion. One of the primary reasons for this was the technical limitations of the CD. While they were capable of greater storage capacity, they impacted performance negatively. This can be seen even to this day, where installing games to the hard drive significantly decreases loading times. For a Nintendo who was looking to bring 3D graphics to the home console, this was a big consideration.

Here is an interview with Saturo Iwata, the former CEO of Nintendo. He discusses the performances issues of CD roms in terms of the movement in Ocarina of Time

Iwata ROM cartridges don't have moving mechanical parts, so you can retrieve motion data in an instant wherever it is, but with a magnetic disk, it takes time to move certain mechanical parts, so depending on where the data is, it takes time to retrieve it, so you couldn't make Link move. If there weren't many movements and you could fit them in the memory, you could read them to memory from the magnetic disk beforehand, but there were 500 patterns.

This quote also points out the consideration of a disk drive system for the N64. This can be seen in the prototype of N64DD, an add on to the N64 posited for release. So while we can look back today and say Nintendo made a mistake, at the time it did not seem this way. In addition to the technological advantages of cartridges, Nintendo had an established cartridge production capacity with its history in the SNES and NES.

Sources:

For basic information about Phillips CDI and Playstation http://ca.ign.com/articles/1998/08/28/history-of-the-playstation

An Engadget article interviewing the Diebolds, who have one of the 200 Nintendo Playstation prototypes https://www.engadget.com/2015/11/06/nintendo-playstation-is-real-and-it-works/

For the interview with Iwata http://iwataasks.nintendo.com/interviews/#/3ds/zelda-ocarina-of-time/1/4

Patent info for the 64DD https://www.google.com/patents/US6769989

There isn't going to be a definitive answer to this due to the nature of the market. Even if we ignore corporate secrecy, you're talking about Nintendo, a multi-national company that had separate offices for their Japanese market and their North American market. So immediately you have two different horses to get information 'straight from the horses mouth.' That said...

Back in the day you basically had one of two choices: Expensive flash memory- cartridges- that had a very fast read time- this meant short or no load screen- or optical media which was cheap, had a lot of physical storage space- 750 megs relative to the N64 carts with had at most 512- but had the drawback of having slower read times which both meant more load screens and that games could lag a bit.

This is the official explanation, at any rate. Nintendo had years of experience with cartridges and had the infrastructure to work with them. A direct quote from Nintendo of America's chairman at the time, Howard Lincoln-

Right now, cartridges offer faster access time and more speed of movement and characters than CDs. So, we'll introduce our new hardware with cartridges. But eventually these problems with CDs will be overcome. When that happens, you'll see Nintendo using CD as the software storage medium for our 64-bit system.

There were, of course, many other reasons as well. Cartridges were fantastic because it made piracy quite a bit more difficult. Proprietary technology meant that unless you somehow got your hands on fairly rare hardware to write to these carts, and then blank carts, you were not going to be able to pirate these games.

You also have to consider that those same proprietary carts meant that Nintendo had a lot more control over their hardware. This was a huge deal in the NES era when Nintendo had to make good on building a, 'we swear we're not Atari, we're not going to single handedly crash the games market' image. At it's height, Atari consoles had a serious issue with quality control stemming from the fact that anyone could make a video game- give two or three nerds a bag of soggy cheese burgers, pay their rent and promise a cut of royalties and in 3 to six months they'd make a game- so virtually everyone did. Even dog food companies had games, and as you can imagine, between the limited hardware and the 'gold rush' on the market, the quality of titles suffered.

Nintendo, on the other hand, was a bit more miserly with who they actually let into the club and they even took cases to court to protect their rights to control their console. Of course on the flip side, Nintendo was also taken to court for unfair treatment of certain companies. If you didn't make your game Nintendo exclusive you might find yourself in a situation where you didn't get as many carts to write your game to as that company next door who did. Of course the same Howard Lincoln rejects this point for the Nintendo 64.

And of course Nintendo was painfully aware of who they marketed their products towards- kids. Without citing a specific issue because I have long since disposed of my copies of Nintendo Power, the durability of Nintendo hardware was a significant marketing point. You go to the Nintendo World Store in New York City and they proudly display an original Game Boy that got torched in the first Iraq War and still runs. You mention durability to Nintendo's marketing department and they tell you stories about dropping Game Boys and SNES's from stories up to see if they'd keep working afterwards. CD's back in the day- although it depended on the materials used to make them- were prone to failures due to scratching. Having a CD meant that your hardware had to have the equipment to read CD's which meant a lot of moving mechanical parts. Flash memory was completely solid state- no moving parts- unless you wanted it to for some odd reason. Nintendo sold their stuff with kids in mind. Cartridges and their respective hardware could tolerate rough and tumble. Sony's stuff and Sega's stuff? Not so much.