Nintendo stuck with cartridges because could make money off of them, and they didn't want to deal with licensing and using CD-ROMs or DVD-ROMs from other companies. That's it. It had nothing to do with load times.
Well, that isn't true. Nintendo managed to get around licensing fees since transitioning to optical media by disabling movie and music playback. It never believed in optical media until technology advanced enough to allow it to not be awful. And Nintendo still talks about the advantages of ROM cartridges. Nintendo didn't stick with cartridges for any one single reason. The reasons were
collectively all the things that have already been discussed.
Nintendo might have gotten away with it if it gave a damn about third parties at all. It never has outside of token attempts there and there, even in the 8-bit and 16-bit days some people look so fondly on. The article isn't available anymore besides snippets here and there, but Emily Roger's "The Sexual History Between Nintendo and Electronic Arts" illustrated this pretty well. It wasn't until Dan Adelman tried to change the culture within Nintendo when it came to indie developers that any headway was made, and he ended up leaving Nintendo about a year and a half ago.
As a company that was homegrown and family owned for decades, it's obvious why Nintendo operates in a bubble. Shigeru Miyamoto created Mario and Donkey Kong because Nintendo couldn't get the rights to Popeye. Then, Universal sued Nintendo for copyright infringement because it cornered the giant gorilla market or something. Next, Sony reached out with a "partnership" that had extremely one-sided terms. Negative dealings with other companies happens enough times and you start getting xenophobic. That doesn't make it right, but it's easy to see where the attitude comes from.
Nintendo seemed to soften its stance on many things once Hiroshi Yamauchi stepped down, but it never lost its desire to remain self-sufficient and independent. That's a different kind of problem though. Sony and Microsoft are overly reliant on third party software. That has gotten so bad that Phil Spencer said in August that Microsoft plans to focus more on first party exclusives. Nintendo's problem seems to be that it thinks it can't do its own thing while also opening the doors and giving back a little.
However, Nintendo has done a better job in recent years, not a good job but a better one. It's licensing out its IPs, making partnerships, and reaching out in unprecedented ways. It's a start, but Nintendo has to get on that. That's my hope for NX. Nintendo doesn't have to stop the good parts of what makes the company what it is to rid itself of the bad parts. Sony is buying exclusives like there's no tomorrow while Nintendo is practically asking not to be bothered. Can it find a middle ground maybe?