The important part here is "if we built". If we built a net-gain fusion reactor our energy problems would be solved too, but we're not doing that.

There are significant problems with breeder reactors and development has largely stopped on them.

The problem here is the AM/FM distinction: Actual Machines vs Fucking Magic.

Fucking Magic is great if you're writing scifi, or trying to sell snake oil to investors. The Hyperloop and FSD are examples of Fucking Magic. Sure, they could, in theory, exist, but they don't, and we don't know how long they would take or even if they make sense in the long term.

There's nothing wrong with working on new technologies that may as well be Fucking Magic until they do become viable.

However, if you are making plans for how to proceed with your policy goals, you need Actual Machines. Actual Machines can't do miracles and fix all of our problems overnight like Fucking Magic can, but they have the benefit of existing. We know their actual benefits and their actual drawbacks. We know that they won't present some brand new problem that makes them impossible to work with, because they are mature. Trains and bicycles are Actual Machines. Wind, solar and hydro power are Actual Machines.

Cars are also Actual Machines, and thanks to over a century of maturity, we can confidently say that they are not sustainable at their current scale. Nuclear fission is similar.

We don't know if Fucking Magic will make the transition to an Actual Machine, and if it does, whether it will turn out to be viable.

If breeder reactors are going to become a technology we can rely on to solve our nuclear fuel and waste issues, then they need to make the transition from Fucking Magic to Actual Machines to finally being viable, and that could take decades or more of further research, and yet more decades to actually build the things. Sure, that could come in time to extend our nuclear fuel reserves before they run out in around a century, but it might not. We just don't know. It certainly won't come in time to make a difference to climate change.