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Following the current trend of pitting the particles に and で against each other, here is another question that does the same but from another type of usage and perspective.
When we want to say "do X before Y", we use "Y 前に X":
食べる前に「いただきます」と言う。
On the other hand, when we want to say "do X after Y", we use "Y 後で X":
食べた後で「ごちそうさまでした」と言う。
What is the simplest explanation to explain the differences between 前 and 後 that make 前 goes with に while 後 goes with で in the two situations above?
で derives from に＋て, and て roughly corresponds to the present/past participles (-ing, -en) in Western languages. Kuno (1973) notices that て implies temporal order. So when you have
走ってころんだ
'(By) running, I fell',
running has to precede falling; it cannot be the other way around. This much is the general consensus.
Notice that the usage of で in the question involves temporal notions rather than locations. Now, I found an interesting explanation here: Q14 that connects the facts mentioned above. According to this, when you have an expression
A [temporal noun] で B,
the て that is included in で obeys the temporal restriction mentioned above; that is, what is expressed by A [temporal noun] has to precede B. Going in the other temporal order is not allowed. Therefore, expressions like
食べた後で「いただきます」と言う。 [Temporal order: 食べた => 言う]
are grammatical but
× 食べる前で「いただきます」と言う。 [Temporal order: 食べる <= 言う]
× 食べるよりも先で「いただきます」と言う。 [Temporal order: 食べる <= 言う]
are not. に can be used by all the examples above:
食べた後に「いただきます」と言う。
食べる前に「いただきます」と言う。
食べるよりも先に「いただきます」と言う。
If there is preference of 食べた後で「いただきます」と言う over 食べた後に「いただきます」と言う, then some kind of slight difference in meaning like what phirru mentions in the comment may be playing a role here.
食べる後でするから is not a correct Japanese sentence.
In 後でする the focus is that you will do whatever you were doing, just later; whereas in 後にする the focus is that you will postpone whatever you are supposed to do until later.
Treating で as the "instrumental" particle, this way, "後でする" would mean "to do (whatever), by using the time after now"
And "後にする" would mean "to do (whatever), at a point of time after now"
So "前に" would be "at a time before now", and "前で" would be impossible since you cannot use a time before now.
Alternatively, treating で as the verb-conjunctive form of the copula だ, ”後でする” would mean "It is afterwards, and do it"
This also shows the impossibility of 前で which would translate strangely into "It is now before and..." which is temporally impossible for the past to exist in the present.
If you want a very simple answer, then you can look at the meaning of ni and de, again: ni means things are separate and interacting. de means that things are contiguous and acting in a similar manner (i.e., to the same ends). I think I answered a question you asked about them earlier to tell you that に and で could be compared to an English analogue of 'each other' and 'themselves'.
Anyway... 後で is used because after the past happens, it is included as a part of a continuing timeline 'themselves'. 前に is used because the prior event interacts with the future event through time or the doer or whatever 'each other'. So, 'ホニャホニャの後でいく' = 'With/'contingent on' whatever's happening, I will do it.' 'ホニャホニャ前にいく' means, 'Whatever happened, and my going before that is related somehow'. And to reiterate one more time: '後で' means: 'the future comes only with complete a specific past' and '前に' means: 'the future seems to interacts with a past'.
I always consider this kind of で as the suspensive form of です. So, I would read the second sentence as "It is the time after lunch; we say gotisousamadesita".
As for why it sounds more natural with 後 than with 前, I have no idea.
的with Japaneseの.