If the convergence thesis is right, why would I own the index instead of the discounted names?

Both things are happening and the split matters more than the headline number.

The aggregate discount is real and not purely an office artifact. Nareit's framing compares equity market valuations to REIT valuations, and that gap has been rivaled only by the global financial crisis and the early pandemic. Office is a small enough slice of the All Equity index by market cap that it can't produce a divergence of that size on its own. Generalist money going into AI names and out of everything else did most of that work, and it hit industrial and residential REITs that have nothing wrong with them.

The separate REIT-to-private-real-estate gap is the longer-running one, the widest stretch since the early 2000s, and that one is closer to a pricing mechanism than a sentiment story. Private marks move slowly. Public prices move daily. When the two disagree for years, one of them is wrong, and public REITs are the side that can transact.

Where your composition worry does bite: if you buy broad, you are buying whatever the index weights happen to be, and those weights already reflect the consensus. Data centers have been bid up on the AI story. Paying up for a structural tailwind and getting a discount only on the parts you didn't want is a genuine outcome of cap weighting.

The thing that decides which action is right is whether supply tightening shows up where you own. New supply is forecast to fall sharply in apartments, industrial, self-storage and senior housing, which is what hands pricing power to existing owners. That is a sector-level fact, not an index-level one.

The risk your framing skips is rates. REITs lean on external capital, so if the easing of rate pressure stalls, the convergence argument can be correct on fundamentals and still not pay for several more years. Timing risk is the cost of being early here, and it is not small.