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Multi-antenna coded caching is known to combine a global caching gain that is proportional to the cumulative cache size found across the network, with an additional spatial multiplexing gain that stems from using multiple transmitting antennas. However, a closer look reveals two severe bottlenecks; the well-known exponential subpacketization bottleneck that dramatically reduces performance when the communicated file sizes are finite, and the considerable optimization complexity of beamforming multicast messages when the SNR is finite. We here present an entirely novel caching scheme, termed \emph{cyclic multi-antenna coded caching}, whose unique structure allows for the resolution of the above bottlenecks in the crucial regime of many transmit antennas. For this regime, where the multiplexing gain can exceed the coding gain, our new algorithm is the first to achieve the exact one-shot linear optimal DoF with a subpacketization complexity that scales only linearly with the number of users, and the first to benefit from a multicasting structure that allows for exploiting uplink-downlink duality in order to yield optimized beamformers ultra-fast. In the end, our novel solution provides excellent performance for networks with finite SNR, finite file sizes, and many users.