Would tung oil normally stand up to that kind of thing?
No. Oil finishes just don't offer the protection that other finishes can (not will, can; it depends on how they're applied) despite any claims to the contrary from manufacturers/marketers and enthusiastic users.
This is because oil finishes are inherently thin. Very, very thin. Most of the buildup (what little there is) occurs in the wood fibres, literally at a microscopic level, with almost no surface film being developed. It is a surface film that imparts resistance to water and stain intrusion, and simultaneously scratch-resistance.
In addition, any plate like this should be washed, just as all wooden items used for direct food contact should be, with the possible exception of bread boards. And this washing will naturally wear the surface down so over time protection will diminish. You've seen this already in a compressed timeframe trying to scrub the stain out, and something like the same rough surface you ended up with will eventually be the lot of the entire plate1.
So as not to sugar-coat this at all, you can see some degradation of the surface of oiled items with a single washing. Add in a realistic reminder that use of eating utensils including just spoons, even wooden ones, will at some point dent or actually scratch the surface and these dents or scratches undermine the finish. So, with an oiled item expect that water will make it through to the surface eventually.
Will it keep resisting stains if the food put on it is warm/hot (up to 140 degrees)?
Heat softens cured oil so you can expect that a pure-tung-oil finish won't respond as well when warm or hot as at room temperature or cold (the softening comes with an expansion, which 'opens up' the cured oil and make it more permeable).
Or did I just not 'cure' the plate long enough?
You may be able to expect slightly better performance after a month or more. But the above comments about the relative weakness of oil finishes will still apply. You need something less permeable, and more of it, to actually impart any real resistance to water and stains.
If tung oil really isn't going to protect the plate from anything warm/hot, is there anything out there that will and is a "natural" product? Don't laugh here
I don't consider it a laughing matter as there are masses of ill-informed peeps out there giving people well-meaning but misguided (and in some cases potentially dangerous2) advice on this issue. On what is or isn't a food-safe issue, the real answer will I'm sure surprise you as it does most people.
See the brief summary in this previous Answer. Flexner has continued to emphasise this point in subsequent pieces.
and the only one that I've found so far that all the "internet experts" seem to agree on is tung oil
You should also find similar widespread, but not universal, agreement on the safety of mineral oil (UK: liquid paraffin), one or other waxes and blends of the two3 such as "spoon butter".
So in case it's not clear you are basically out of luck here, you're seeking a unicorn finish. There is nothing that most people would find acceptable that will give a close-to-the wood appearance (no surface film, which looks and feels most natural) while providing good or excellent resistance to water and stains.... if there were I suggest it would already be in widespread or nearly universal use!
Just to mention in closing, wooden trenchers and other food-service items would often (perhaps nearly always) have been used bare. In many cases this would have been done for no other reason than economy, but it was likely found through experience that it wasn't necessary anyway. They wouldn't have cared much about staining.
1 Pre-raising the grain during the final stages of surface prep can diminish or remove the potential for the grain to rise when the wood does eventually get wet again.
2 For example, mineral oil is a suitable finish for food-prep and food-service items and it protects the wood and, often as a following point, keeps it clean. It's the last bit that is potentially dangerous as some people are going to take that ball and run too far with it by assuming that coating a piece with mineral oil means they don't have to wash it.
3 My opinion only: you should experiment with waxes, especially applied hot or the entire item dipped in hot wax where feasible. Ignore wax/oil blends as a dead end, blending wax with oil just weakens the protection provided by the wax.