It's not really that strange; the M16 and it's variants have been in use for 53 years. Mosin-Nagant rifles were first used in 1891 and still serve as discount weapons for various paramilitary groups - there are recent photos from Ukraine, which is ironic because Ukraine was ruled by the Russian Empire under Alexander III when that was first introduced.

The Brown Bess had few technical advantages besides being so simple that there's very little that can go wrong with the gun itself. The Ferguson rifle of 1775 was a breech-loading rifle that was technically better than any muzzle-loading smoothbore by a landslide; six shots a minute instead of two and a range of 250 yards instead of 50. However, it broke frequently and was extremely challenging for primitive gunsmiths to produce. Only percussion caps and advances in machining could kill the Brown Bess pattern, and after those advancements were made it wasn't much further to cartridge bolt-action rifles such as the Lee-Enfield and the Mosin Nagant.

Development of the Land Pattern musket was not completely stagnant, there were 5 variants developed with efforts made to adapt the weapon for use on horseback or at sea or to reduce the cost of production. This final point is very important - one of the great successes of the pattern was that it was designed for mass production and so large numbers of weapons could be produced relatively cheaply and to a certain basic quality standard.

Attempts were made to find replacements for the Land Pattern, but the sheer expense of the Napoleonic Wars forced the Ordnance Board to stick with it. In British Military Longarms 1715-1865, De Witt Bailey lists a number of of potential competitors to the Land Pattern including the "Duke of Richmond's Musket" which he describes as "the highest degree of perfection attained by the British Military flintlock". Alas this musket was developed in 1798 and could not be afforded for mass issue, although a few made their way to militia units.

In short the Land Pattern was cheap, reliable and as effective as its counterparts so there was no need to do away with it.