Competition would be wonderful, but is simply not practical in many
cases. Most people and companies don't want to hear this, but from
a consumer perspective the Internet is a utility, and very closely
resembles water/sewer/electric/gas service. That is, having 20
people run fiber past your home when you're only going to buy from
one of them makes no economic sense. Indeed, we probably wouldn't
have both cable and DSL service if those were both to the home for
other reasons already.
The copper pair from your house to the exchange isn't congested (at least, between you and other people), by definition.
The fibre from your house to somewhere isn't congested, in the same way - although the 'somewhere' may be closer to you than the exchange.
There's no reason to have competition here - but there's no reason to have a commercial entity trying to make a profit here either. Treat it like a real, basic utility, and run it on a cost-recovery basis, either directly by your local government entity, or by a dedicated organisation acting on their behalf. Whatever entity owns the local-loop sells access to competing service providers on an equal and transparent basis.
Providers can then compete on level of network congestion, amongst other things, because they can directly control it in terms of how much backhaul they want to build from the exchange / FTTP street cab / whatever the aggregation point is against the volume of subscribers they sell off that aggregation point.
For places which have it, this seems to work much better, or least break far less, than the alternatives (Openreach in the UK, Stokab in Stockholm being two I've dealt with).
Exactly like electricity and gas - I only have one set of wires / pipes to my house, but there's a plethora of companies I can choose to buy energy services from.