Well, I graduated many years ago, not sure the below experience still relevant.

I had my internship in one of the well known mid-tier firm before, and I joined one of the big 4 once I graduated from university. During my internship days, I was assigned to perform P&L testing, simple analytical procedures, review of statutory records and simple test of controls (SME engagement). My friend who had internship in one of the big four, he was assigned to perform P&L vouching (MNC and listed company audit).

I believe that's the exposure part mentioned by your lecturers. Nevertheless, I don't think the internship exposure matters much for your future employment, since an intern will not be expected to involve significantly in the audit engagement.

However, if you would like to get into big 4 in the future, maybe you should just apply for big 4 internship program. If you get to know some managers/seniors, you may get a referral in the future right.

Just my 2 cents.

From my own perspective, MBA is not relevant for the audit/tax/advisory role in Big 4.
Big 4 do take in ACCA affiliate / degree holder, some even take in non-accounting graduate, such as PwC KL.
Even you would be able to join Big 4, it's usually the case that your peers will continue to pursue professional qualification, such as ACCA/CPA Australia/ICAEW.
If you spend time on the exam techniques in answering ACCA questions, it is not difficult to get 50 marks though.
Anyway, wish you luck to get into Big 4.