The tunnel in Southampton is a bottleneck, yes.
The reason that it's easy to terminate things at Southampton is not just that there are four platforms, but there are four reversible platforms. So unless there's something in every platform there will always be a line free for something else (such as the two freight trains an hour from the Midlands and North). Also it helps that the platforms at Southampton are divided, so you can terminate two services in one platform.
Winchester only has two platforms, and they are not reversible. So if you have a terminating train on the up platform for a few minutes, it means that no trains can pass in the up direction. Which is a problem.
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EDIT: Okay I've looked it up, and it turns out the lines through Winchester are bidirectional (although the speed limit is a lot lower for going the wrong way), but there's a bigger problem: there is no crossover south of the station. The next crossover to the south is Albrook Junction, over six miles to the south. So anything terminating in a platform will still block the line, because it's obviously not possible to have all other services travel wrong-line for six and a half miles.