A CBDC by definition isn't "split between multiple institutions": it's emitted by a single central bank, which doesn't need or want a mechanism to obtain "consensus" of third party users of its currency (institutional or otherwise) for its operational decisions.
Similarly, it doesn't have any need to invent Lightning networks to overcome the crippling limitations some blockchains impose on transaction throughput (and it's not like you need blockchains to implement payment layers and eventual consistency of records where that's seen as preferable to instant settlement) and if central banks want people to be able to write 'contracts' to automate interactions with its money to a greater extent than is already possible through third party institutions, it's perfectly capable of providing a permissioned read/write API to its records without implementing itself on the Ethereum blockchain.
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A CBDC by definition isn't "split between multiple institutions": it's emitted by a single central bank, which doesn't need or want a mechanism to obtain "consensus" of third party users of its currency (institutional or otherwise) for its operational decisions.
Similarly, it doesn't have any need to invent Lightning networks to overcome the crippling limitations some blockchains impose on transaction throughput (and it's not like you need blockchains to implement payment layers and eventual consistency of records where that's seen as preferable to instant settlement) and if central banks want people to be able to write 'contracts' to automate interactions with its money to a greater extent than is already possible through third party institutions, it's perfectly capable of providing a permissioned read/write API to its records without implementing itself on the Ethereum blockchain.