There are various ways you can be covered while riding a bike -- glad no injuries. In some states you are covered under your auto policy for medical -- it gets complicated. Don't delay with seeking medical if any symptoms pop up.
As to the bike ... Anothers liability insurance is not direct to you. They don't have any contract with you -- they pay based on a judgment. Most companies are not going to wait for this to occur when the insured is clearly at fault and they would lose in court. But -- its really a situation of now having to show or prove damage to the bike.
Getting something in writing from a legitimate source that a carbon bike cannot be guaranteed safe after a crash would be wise to obtain -- if that is possible.
You would have the same issues under a homeowners policy .... even with that being first person. There has to be proof of a loss ...IE damage. That would also be subject to any deductible. Also, many standard mass market homeowners policies are full of sub-catagory loss caps. IE -- we will pay to replace bicycles at value up to a max of $1500. So you can have three bikes each worth 5k and you get $1500 for all of them in a house fire.