The improvement I'm suggesting is to account for the nominal rigidity of rents by looking at current market prices instead of average prices. The current practice is to survey households about their current rent expenditures, but most of these expenditures are based on leases that reflect prices which are set in the past. This is kind of (though obviously not entirely) like looking at changes in average household mortgage payments to gauge current housing market prices. A measurement based on market rents, in contrast, would reflect the prevailing market rents that are charged, and would therefore better reflect current housing market conditions.
We don't gather food price statistics by asking people what they paid for the goods in their pantry; we go look at what those goods are selling for on the market today. Seems weird that we don't do the same for rent prices.