Sometimes, about once every 15 launches the game won't start at all. I only get the Red Mojang screen, forever.

When that happens, a rare fix can be to restart, but this takes a lot of attempts. It doesn't seem to matter whether you restart the console or just the game. It can easily take dozens of attempts. A much faster fix is to reinstall Minecraft. It isn't fast, but at least has a 100% success rate. (odd side effect: In roughly 25% of cases the connection with the Microsoft account is lost and I need to login again.)
Yes, thisis happening so much that I can have statistics on the problem.

It started with the 2.07 release, and at the time it was a worldwide problem. This was fixed quickly, but hasn't fully gone away for me. Still happens with yesterdays 2.15 release (these are the version numbers mentioned on the PS4)

I have the impression the issue is more likely to occur when turning off the PS4 without closing Minecraft, but it does also happen when Minecraft was properly closed before turning of the PS4. Maybe it happens more often after a non PS+ account has played (that was the person who shut down the PS4 without quitting the game, or even the map. I'm not sure. I have not been able to deliberately trigger the problem, it could just as well be random.

I have first bought Minecraft as digital download at the PS3. I found people reporting it could be a license problem and rebuilding the database could help. Not yet tried that.

There are 3 players on my PS4 who regularly play. One with PS+. We regularly play local and online multiplayer. Not sure if this matters.

I have one payed pack: Farmcraft. (I think I bought this when the problem was already occurring, so I don't think that matters) and I noticed that every first time you play that game in a Minecraft session, it downloads all packs, as if closing Minecraft also removes the Farmcraft packs. I don't know if that's normal.

Yesterday, after the 2.15 update I started Minecraft, played local, single player for less then a minute, exited the map,  and closed Minecraft and a few hours later put the PS4 in standby. This morning I started Minecraft, got stuck at the red screen and gave up after 20 minutes. I restarted the Playstation and started Minecraft, got stuck at the red screen and gave up after 10 minutes. I then reinstalled Minecraft and the game started normally.

So even with the latest version, doing little more than launching the game, the problem still occurs.

I have never experienced a similar problem with any other game. It only happens with Minecraft and it happens often.