"If broadband connection drops (and mine appears to do so quite frequently) what would be the point, if any, in having a router?"
For the purposes of this discussion the difference between a modem and a router is irrelevant, all you need to know is that ADSL routers are an ADSL modem with a couple of other bits in the box that enable more than one computer to be connected at once (and a router may, or may not, include a wireless access point)
Like most things, not all ADSL modems are equal (regardless of if the modem is a stand alone one or hidden inside a router) I've seen many people have serious problems with ADSL connections that have been fixed by simply user a better quality modem/router.
There are, however, a few things which effect the quality, speed and reliability of an ADSL connection.
Distance from the telephone exchange.
I live 600metres from the exchange I'm on, I get a reliable connection of 6-7Mb/s and it often peaks to over 8Mb/s, My parents live 2-3miles from the exchange in their town and they only get 1.5-2Mb/s
Quality of your houses internal telephone wiring (if not connected to the master socket)
A lot of cheap telephone extension kits are made using the cheapest cable the manufacturers can source, this cable is fine for voice telephone calls, it's not up to the job for the higher frequencies that ADSL uses.
The ADSL filter
The free filters supplied by your ISP are the cheapest ones they can source, they are not high quality ones at all and can seriously effect the reliability of your connection, they also do go wrong quite often.
Your modem (either stand alone or combined inside a router)
As said above, not all modems are equal, I've seen many cases of bad connections on cheap (often ISP supplied) modems and routers that simply replacing them with a £30-40 netgear router (the best routers at hanging onto low qualit