It's not an either / or. Most portfolios will/should have both.
In some investment scenarios Index ETFs are a good choice - eg for a broad based Large Cap investment the Nifty50 offers great diversification at minimal cost.
If you want a narrower focus, the Nifty 50 Value 20.
If you want a broad sectoral bet - the Bank Nifty, or the CPSE 22.
Etc You get the point.
An active mutual fund primarily solves the discovery challenge retail investors have. When do you need this the most - when you're fishing in a largish pond where formation is difficult to come by - eg Mid Caps, Small Caps, Micro Caps.
There of course are Large Cap MFs. They work off narrower portfolios and active management that "could" yield higher returns than the Index. Costs are proportionally higher vs ETFs.
In summary, choose horses for courses. Figure out a core/satellite strategy and the choose instruments. Not vice versa. Also run investments and bake portfolios focussed on goals, not which product to choose.