Sean, the AIP entry for LHR mentions...
"Descent profile: On receipt of descent clearance the pilot should descend at the rate he judges will be best suited to the achievement of continuous descent, the objective being to join the glide path at the appropriate height for the distance without recourse to level flight."
The aim at LHR is not to have aircraft fly level for a mile or two under the GP before descending, but to arrive on the LOC at or slightly above the GP and continue descending without leveling.
Low power/low drag CDAs are the order of the day.
Better use should be made of the already available phraseology of "when LOC established, descend on the glidepath", it can be used on the closing heading, or the perhaps not quite MATS Pt 1 "descend to x thousand feet then further on the glidepath".
Making use of the LCY departure display that LHR FIN has available to them, I also see no reason why descent below 4,000ft cannot be given a little outside 13nm either in order to keep a CDA going as appropriate.