score:5
Timatic was actually correct (at least, more correct than EU). The new Hungarian classification system will come in effect only on July 15. The EU website now also notes this.
Additionally, the EU website is really not focused on entry requirements generally, and its wording is biased towards assuming readers are EU nationals in EU.
For example, it says
Travellers can enter Hungary without any restrictions from countries belonging to the green category.
But in the other tab
As a third-country national coming from outside the EU and Norway, Switzerland, Iceland and Liechtenstein, may I enter this country without exceptional restrictions?
For third-country nationals the entry into the territory of Hungary is not allowed, except for humanitarian purposes or in the following cases: third country citizens which are familiy members of an EU or EEA citizen, citizens of Serbia, nationals of Ukraine for a maximum period of 24 hours and within 30 km of the border area. Special rules are established for citizens of Japan and South-Korea for business acitivities purpose.
do they have access to more up-to-date, exhaustive information, e.g. coming straight for government officials without any Timatic delay?
At least some airlines do receive instructions from aviation and immigration authorities directly, especially for airlines that focus on a specific country. But the information is far from exhaustive.
e.g. Chinese airlines do receive information from CAAC directly on China's restrictions. They probably receive information from some of the other countries they serve as well, but may not want to spend the labour and time to process all of that.
Also, employees that welcome you at the airport are not necessarily employees of the airline, and airlines may not be able to communicate information efficiently to the frontline. Transit complicates things more.
If they do, can passengers have access to some/all of them so that they can better estimate their odds of reaching their destination and plan for traveling?
Ask. Ask the immigration authority and the airline (for transit points as well). Not that they would always understand you fully and give the correct information, but that's the only thing you could do. Airlines are also more willing to enforce restrictions than to allow exceptions or relaxations.
Written confirmations from immigration authority are helpful. Be assertive (but respectful) as well (if you are sure you are right, of course.)
Usually, many government regulations are drafted, released for public comments, and then published officially before its coming into force, with plenty of time for airlines and Timatic to prepare and update. But under current exceptional circumstances, you can't be sure until you go to the airport on the day of departure.