You are confusing different kinds of というのだ/っていうんだ. The one you said you know it's an exclamation marker is like the followings.
分かってるっていうんだ。
いつ来るっていうんだ。
早く来いっていうんだ。
In this case, the clause preceding と/って is the speaker's words (including question or imperative), and the subject of 言う is the speaker. Indeed っていうんだ can be omitted without changing the meaning here.
分かってる!
いつ来る!?
早く来い!
一体春休みに何があったっていうんだ! is also this structure, so 一体春休みに何があった!? has the same meaning (一体春休みに何があったんだ is more natural, though).
On the other hand, in どうしろというのだ in the thread title, the clause preceding と, i.e., どうしろ is the contents of the addressee's words (request/order), and the subject of 言う is the addressee. This is actually a rhetorical question, so the actual meaning is どうにもできない/どうしようもない. When there is no interrogative in the clause preceding と, というのか is used.
何を食べろというんだ(=何も食べられない/食べるものがない)
どこに行けというんだ(=どこにも行けない/行くところがない)
これを食べろというのか(=これは食べられない)
あそこに行けというのか(=あそこには行けない)
In this meaning, というのだ/か can't be omitted.
There also are other usages of というのだ, and the meaning/nuance can be different depending on the context or intonation. You need to keep in mind that it's not that simple.
if the meaning is "you said ...", why don't we say といったのだ?
It's the same reason as why 信じられん can be used there instead of 信じられなかった. It's not uncommon to use the present form for the past event in Japanese. You must have already come across so many examples, I believe.