The link to the feature provided here explains how it works. Your Bitwarden client has a master key that opens your vault. This is your own private key. Bitwarden the server doesn't have a copy of it. If you choose to designate a trusted successor in the case of your incapacitation, you send a request to that person. If they accept the request, Bitwarden the client will generate a public/private key pair for them. They keep the private key in their client. Bitwarden the server gets a copy of their public key and sends it to you. Now your Bitwarden client encrypts your private key with their public key and Bitwarden the server gets a copy of that. In the event of your incapacitation, they send a request to get a copy of that encrypted key. After the timeout period, Bitwarden the server will send your encrypted master key to them. Then their copy of Bitwarden the client, which has their private key, can decrypt it, and now they have a copy of your private key as well.
At no point does Bitwarden the server have a copy of anyone's private key. And no splitting of keys is necessary. This is just the normal way asymmetric encryption works.
This, of course, all breaks down if you don't trust Bitwarden the company, since they provide you the client. As far as I understand, US law enforcement doesn't have the legal ability to force a company to modify their own software to make it malicious (as opposed to doing something much simpler like forcing them to turn on IP logging on a VPN server). But if your threat model includes the possibility of US covert intelligence services MITM-ing Bitwarden the company and sending you their own malicious client, then yeah, keep your secrets in a physical vault guarded by people willing to die in a shootout with the FBI before betraying you. Make sure they'll answer to your successor if you die.
Ok then, so one search warrant to Bitwarden to get the encrypted key and your encrypted passwords, and another to the authorized party to get their private key.
It still comprises a break in the end-to-end crypto, and can still bypass the time delay and decrypt your passwords today without your involvement.
At no point does Bitwarden the server have a copy of anyone's private key. And no splitting of keys is necessary. This is just the normal way asymmetric encryption works.
This, of course, all breaks down if you don't trust Bitwarden the company, since they provide you the client. As far as I understand, US law enforcement doesn't have the legal ability to force a company to modify their own software to make it malicious (as opposed to doing something much simpler like forcing them to turn on IP logging on a VPN server). But if your threat model includes the possibility of US covert intelligence services MITM-ing Bitwarden the company and sending you their own malicious client, then yeah, keep your secrets in a physical vault guarded by people willing to die in a shootout with the FBI before betraying you. Make sure they'll answer to your successor if you die.