The one time we went back we travelled on their British passports and nothing was mentioned. So, just because they were born in the USA they have to own a passport?
Just because nothing was mentioned that one time doesn't mean you won't have issues in the future. If the US immigration officer had realised they were US citizens trying to enter on a foreign passport, your children could have been refused entry to the US.
They need US passports to enter the US because they are US citizens. Whether they were born in the US or if they had been born in the UK to a US-citizen parent, they are still US citizens and must therefore use a US passport to enter and leave the US.
If you will never again take your children to the US, then you don't necessarily need US passports for them - but if they will be entering the US at any time in the future, they must hold a valid US passport in order to do so.
It's an added expense which will hardly be used. We do not have a reason to go back anymore.
I read that they could travel on their British passports and take their expired US passport if they needed a proof of nationality. But we consider their nationality to be be British.
Whether you regard their nationality to be solely British or not is irrelevant. The fact is that they have dual nationality and are citizens of both the US and the UK... which means they have to abide by the immigration rules of both countries.
If they want to enter the US in future, they must travel on a US passport. That's the law.
I'm not sure whether an expired US passport would be sufficient or not... however, as US citizens, they also cannot legally hold a US visa or ESTA, so if they tried to enter the US with a UK passport and ESTA, that could be a problem as well.
So, if you don't want to get new US passports for them right now, that's fine, but you will need to make sure your children do not attempt to enter the US again without applying for US passports before they travel.
Further information on how the US views dual nationality can be found here:
https://travel.state.gov/content/travel/en/legal/travel-legal-considerations/Advice-about-Possible-Loss-of-US-Nationality-Dual-Nationality/Dual-Nationality.html