The example that you identify as correct expresses a general of habitual state: John likes ice cream. This is not something John did yesterday, or at any specific time, it expresses a general truth about John's likings.
This is one of the ways in which the simple present is used, and when used in that way, it can be used in a subordinate clause following a past tense in the main clause:
Yesterday I found out that Peter is married.
Last year she told me she is an avid bird-watcher.
Last week I asked him if he enjoys swimming.
However, semantically, some things cannot be parsed as a general state or a habit. If I tell you that I am hungry, it is unlikely that you understand that I am a kind of person that is usually or always hungry, or that I am in a continuously hungry state. So if I am hungry now, that implies I want to eat something and stop being hungry.
That is why this sentence feels funny:
A family was celebrating a wedding when somehow a rumor spread that bride is feeling hungry.
Unless we are to understand that the bride is "a hungry person", the sentence is confusing. If we do want to make a statement about the bride's eating habits, we could say:
A family was celebrating a wedding when somehow a rumor spread that bride loves fine dining.
In case of the bank being in debt, something similar is going on. We don't interpret "the bank is in debt" as a habitual or general state for a bank (although that may actually be changing the in light of the last couple of crises …). If we want to express a general feature of that bank, we could say:
A year back there was a rumour that Citibank wasts a lot of money.
Now, even in the sentences where we can use the simple present to express a general property of habit of the subject, it is very common to still use the past, in line with the main clause. This does sometimes raise question (here and on ELL) as to whether it shouldn't be the present if the described property still persists at the moment of reporting:
A rumour spread that the bride loved fine dining.
I found out Peter was married.
Although we will generally understand that the bride still loves fine dining and Peter (in absence of proof to the contrary) is still married, some people may ask if those things still hold true today.
