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In that sense, empathy is a way to feel good about your own ability to manipulate your personal experience in the way how other people must be feeling.

This assumes you're operating under the assumption that you have a mental model for the other person. This is not necessary, you can be open to others feeling and thinking in very different ways than you are, and using an imperfect medium (speech) to best understand it.

Not knowing with 100% accuracy that your experience matches does not invalidate the common framework that myself and another human are operating on. You can say with reasonable certainty that a computer performing a calculation is doing so very similarly to another computer. The same is true of humans because we share similar 'hardware'.

To simplify this down to 'we cannot ever feel empathy, only sympathy' can be a useful exercise to frame how little we truly do know, but to make the jump to 'therefore everything must be framed through your own eyes' is ignoring the shared biology we have. There are behaviors that we know humans will spontaneously show, and there have been countless discussions where ideas have some level of universality to them. To ignore these is an oversimplification and shows the error of this level of inference.

As a short aside, the model you are working with 'you simply looking for pleasurable extension of yourself in other people' is just proof of that error of inference. You yourself already stated that you cannot exist in the mind of others, thus it is plausible (and likely) that someone else exists which does not conform to the same model that you have proposed. Compersion is a good example of a kind of behavior which does not fit your statement.