Better still, abandon CPI and CPIH as headline series and concentrate on the Leyland/Astin methodology.
I largely agree with you. For a traditional consumer price series, John and Jill's broadly defined payments approach is the best. I sent a note in that vein on January 17, or tried to, but it seems to have disappeared in a galactic worm hole, so I am resending it below. Whether it should be the headline index depends on whether you think the traditional consumer price series or the macroeconomic inflation indicator (a reformed CPI, but definitely NOT the existing CPIH) should be the headline measure. There is a lot to be said for making the macroeconomic index the headline measure, but reasonable people can agree to disagree on this.
Just to support Shaun's point, although there is no owner-occupied housing component published for the Retail Prices Index a weighted average of the OOH-related components of the RPI would indicate an inflation rate for OOH costs of 3.6%, or 2.9% if you make the adjustment for the formula effect:
Dec 2017 annual inflation rate
Mortgage interest payments
Repairs and maintenance charges
Dwelling insurance and ground rent
Estate agents' fees
OOH components of RPI
Note that even more than housing prices reflected in depreciation, dwelling insurance prices rose strongly. My rough estimate includes an assumed expenditure weight and an imputed price increase for estate agents' fees, part of the RPI for could be improved if it incorporated estate agents' fees, part of the RPI for fees and subscriptions. ONS staff could certainly calculate a better estimate than I have done, but it wouldn't be very different from mine. The RPI treats stamp duty as out-of-scope, and its inclusion might also bring the OOH inflation rate up.
The biggest claim for a rental equivalence approach to OOH is its consistency with the System of National Accounts so the CPIH doesn't and logically can't include estate agents' fees and stamp duty since these are part of gross fixed capital formation in a national accounting framework. Most of the other components are supposedly covered off in the imputed rents series although obviously not in a satisfactory way if the CPIH imputed rent series is so much lower.
In the long-term a household inflation index as recommended by John Astin and Jill Leyland represents the best measure of the increase in OOH costs. Unfortunately, the HCIs that ONS have calculated so far don't properly reflect their vision, treating both equity payments on houses and renovation expenditures as out of scope. So for the moment, the RPI, living on life support, with its notorious formula bias, remains the only official price series designed for traditional purposes that is appropriately sensitive to housing prices.