So, if we can get pet births down to replacement levels, problem solved?

Alfie looking for a home

Not entirely.

It's quite possible to have a situation where most of the young animals are wanted but there's still a substantial welfare issue about animals being discarded when they don't match owners' expectations.

It may be possible to solve this partially by a combination of education, help for otherwise good owners who run into financial difficulties and improvement of the conditions in which dogs are reared so that fewer end up with behaviour issues. However there will always be some people who die, get ill, lose their jobs etc. etc. and can't keep their animals for reasons that are not their fault. Barring an unlikely amount of improvement in human behaviour there will probably always be some owners who aren't suitable and a few who are actively cruel and whose animals must be removed.

So long as there are more people looking for animals than there are animals needing homes, this just means that animal shelters will still be needed, but most of their occupants will fairly readily be placed. Unfortunately it doesn't mean this will apply to all animals taken in.

Some animals will have chronic health conditions that make them less attractive to potential adopters (or simply unaffordable because of the cost of treatment). Some will be elderly. This doesn't mean it's going to be impossible to save them, just that the simplistic model of "find good home, problem solved" won't work. Realistically the majority of adopters want a pet with good prospects of being a companion they'll have for many years and few are going to consider a relationship they know will end in heartache in a few months or years.

This means high-quality, long-term fostering is likely to become more important to avoid elderly or difficult animals simply being warehoused in kennels.  Getting it right will be crucial, so that no-one is overwhelmed with the stress of looking after multiple animals who are approaching the end of their lives. Cost isn't the whole story; it's the cumulative load of animals needing to be taken to the vet nearly every week, tempted to eat and helped with grooming.