After some investigation and benchmarking, it looks like the best PIR protocol for this use case is YPIR+SP (from February). On a single compute- and network-constrained server, with users on constrained (and possibly metered) networks, this would amount to providing service to up to 1000 users while keeping latencies reasonable; by (quadratically) scaling the server(s) enough, that could become up to 100,000. That means this method of message routing could definitely work, although I look every day in case new protocols are published.
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Sorry! I meant Private Information Retrieval, that could allow metadata-hiding messaging.