@MarMarcin It's possible that one of the game patches broke the omni plant opportunity, or made it more likely to break, although I don't know for sure. (By the time I patched to 1.63, I had been using mods for a long time.) There are a number of in-game functions that broke because of a patch and were never fixed.
Yes, you could, in theory, add a few mods to the game, use them to fulfill the opportunities you want, and then remove the mods again. But you'll probably see more errors later, which the same mods could correct. A good example is the borked adventure board you're seeing in China and France. I think it was never reliable without mods, but it probably got worse as more patches were added to the game. NRaas Traveler fixes many aspects of vacation and travel, including the adventure board, and it also makes the travel transition itself much more reliable, so that your sims are almost guaranteed to arrive safely.
I understand that many people are reluctant to use mods, and there are many mods that introduce extra sources of error. But this is not true of NRaas mods—they are specifically designed to fix bugs and glitches, clean up bad game data, and generally keep your saves healthy for much longer than they would be otherwise. The main cleanup mods are ErrorTrap and Overwatch, and they and Traveler don't affect the way you play. They do their job in the background, without your needing to issue commands. Other than the occasional report of an error that one of those mods has cleaned up, you would never know they were there.
I should also say that unlike some mods that have special uninstall instructions, and others that can't be removed safely at all, the mods from NRaas can be added and subtracted from your game whenever you want. As long as you delete the scriptCache.package file, you'll be fine.
Normally I wouldn't recommend mods to someone who is so reluctant to use them, but it sounds like you're having a lot of issues that could be easily addressed. And there's no way to do so just in the vanilla game—EA never bothered to fix any of the problems you've mentioned. Without mods, getting around the bugs is mostly a question of luck, and trying over and over again until something works.
One other thing to mention is that if you want to use mods, you'll need to patch to 1.67. It's easy enough to do—just download and run the Super Patch. You don't need to patch to 1.69 for any mods to work, and that won't fix any game issues, either; the only "updates" from 1.67 were the mandatory Origin tie-in and the pack selector. But even if you don't use mods at all, it's still a good idea to patch to 1.67. A few issues did get fixed, after all.
https://forums.thesims.com/en_US/discussion/175477/official-ea-download-link-for-the-sims-3-cumulative-manual-update
Just be sure to delete the five cache files in your game folder in Documents after you patch.