Why can’t I complete the Rangfust puzzle in the Time Machine level

To get the achievement, you just have to make sure you have the postcard in your possession at the end of the game. You can set the postcard down at any time, as long as you pick it back up before you leave an area. (Otherwise some puzzles would be impossible because you need all three characters to hold a different item at the same time.)

However, for the trick you suggest (leaving it in the lobby at the beginning of the game): my guess would be that it would not be there when you return to the gift shop at the end of the game. I am going to do my second play-through soon, so I will try it and let you know.

It is not really a hard achievement if you still plan to play through the game again with other characters. It would just be a pain in the neck if you have already played through the game with all the characters because you would have to do another play-through just to get it the trophy.

Update: I tried your suggestion on my second play-through. I left the postcard on the floor of the lobby at the beginning of the game. At the end of the game it was not there still. I think you have to lug it with you.

That's fine. As with the other approach, bring the boat through those same places.

You should realize that you can push the boat from behind so just push it into the now emptied area where the door switch is located. Water or not, the boat should fall so you can continue solving the rest of the island.

On some of my several playthroughs to get all achievements, I did it in this order. It's even done in this order in this guide.

There are no dead-end puzzles that I've encountered in this game and I've tried to find them. I've managed to come up with some cheap solutions thanks to time-traveller's teleportation, but never been stuck.