I can't see this as criticism of hedonism. If anything, favoring being in the moment is moving us closer to hedonism, splitting the difference between religiosity and hedonism: We are not obsessing over duties, but we are not looking forward to the next treat, either. So, if this is a dig at someone, it is Christianity, which, by this logic actively undermines its own stated goal, by giving people explicit, unnecessary things to think about, and placing too many of those things in the future. Hedonism is at least consistent.
Wittgenstein seemed to many to want to maintain and broaden what is currently our "mystical" religious sensitivity, but extricate it from religion. I think this is a use of logic in the service of practical mysticism.
To some degree, I think he is arguing for pursuing a stereotype of Zen Mushin or the Chinese Taoist 'doing in not doing' thinking. This is not alien to a Western perspective, and shows up in mystical exhortations like "The Cloud of Unknowing", is an aspect of many uses of both the rosary and the 'Jesus Prayer', and has been adopted by un-ministered Christian sects, like Quaker meetings, as the basic form of worship.
If you want eternal life, where 'eternal' does not mean 'forever', but 'beyond time', then future-focussed cultures like ours have the whole approach wrong. Eternal life simply does not mean immortality, which is impossible. It means putting aside time.
One should not focus on the future, and God's future judgment of Mankind, or on your own state after death, but should instead learn to step outside of time in the present. People naturally do this when they suspend concern about the past or future. So we often all have eternal life, in the form of 'flow' or other partial suspensions of consciousness. We just don't value it. And unless we cultivate it, we find it elusive.
One part of the equanimity one pursues in mastering that state, in many traditions that teach it, is the awareness that your own death is a specific, unique event only for other people, and not for you. After all, by their logic, when your consciousness ceases, it should be very much like other times when you have escaped explicit conscious thought. This time, you just won't be returning back to consciousness. Taking away its uniqueness makes it more approachable and less scary.
Psychologically, we pay a price for not valuing and cultivating this state of mind. Practitioners of Dialectical Behavioral Therapy find the ability to regulate emotion is improved by undirected meditation over time. And various hallucinogens that purposely shut down or thwart different aspects of logical thought have been shown to work similar changes.