In analog television, a new video field (or a half frame) was guaranteed to be received in 1/60th (or 1/50th for PAL or SECAM) of a second once the tuner locked in to the new channel. But with digital TV, data have to be demodulated, demultiplexed, decompressed and decoded, all of which takes finite time.
You might experience a miniscule improvement in channel switching time employing multiple tuners (instead of a single tuner implementation) in a PVR (where the content displayed is actually the immediate playback of the ongoing recording). The PVR would have to intentionally optimize the switchover to initiate the new recording & playback in favor of shutting down the previous channel. But actual PVRs do not utilize their tuners that way.
But in general, multiple tuners could not offer faster channel switching simply because, once you specify the new channel, there are latencies in demuxing the Transport Stream, acquiring the MPEG streams and decoding them prior to display. There may be latency in simply receiving the packets for the subchannel you selected; i.e. digital data uses time multiplexing rather than frequency multpexing of analog transmissions. The MPEG decoder may need several GOPs (group of pictures) consisting of I, B and P frames before video can be rendered. The broadcaster can make this channel-switching latency worse by employing a large MPEG GOP length combined with a large number of sub-channels. (A few of my local channels require as much as 10 seconds for video to appear, although audio may take only 2 seconds and a premature image might appear at 9 seconds but have inverse luminance.)
Multi-tuner cards are intended for concurrent recording of different channels, and/or watching one channel while recording (an)other channel(s). Channel surfing will typically utilize a single tuner (for conservation of resources).
The need for splitters would depend on how many tuner inputs (F-type connectors) there were on the multi-tuner card. If there was only one input (and the splitter was internal to the card), then obviously another is not necessary. But some multi-tuner cards/boxes have multiple inputs, sometimes an individual input per tuner for maximum versatility (e.g. some tuners for OTA, others for cableTV or different antenna). If there are multiple inputs but only one source, then you would need a splitter or a distribution amp to feed all the inputs/tuners.