Is that the real reason? Being bundled with the OS as the default browser for 95% of all computers seems like a bigger driver!

Neither of those are the reason, speaking from experience both as a family tech support and as a web dev. It’s because everyone recognized the big “e” icon. My brother got my dad a Mac and he already knew the main browser was Safari, but he still called me (knowing I had used Macs for years longer) and just asked:

1. What the hell is Safari (no kidding he didn’t even say hello).

2. Can I use the Internet Explorer?

Edge now (correctly) accounts for this by... having a big “e”-looking icon and basically otherwise being a skin over Chromium.

I think you're confusing cause and effect here.

Yes, the `e` became synonymous with the Internet for a lot of users, but that's only because of the two reasons listed. Not the other way around.

Internet Explorer became popular because it was bundled with Windows.

At the time, IE was also dramatically faster than Netscape Navigator, because it wasn't just bundled, it was built in.

Was it really faster? I think IE had faster start-up time, since some of the components were loaded along with Windows boot, but I don't think it was faster overall.