I mean it depends on the religion. Some are more 'enlightened' than others. Ultimately I find that often time people hold values that have little to do with the values of their religion but they just decide, despite evidence to the contrary, that this is what it means to be a good -insert religion-. A lot of what people think make them good -insert religion- is just , for lack of a better term, due to the westernization of their cultures. Which often leads to hilarity as the same people then profess dislike of the west most of their moral values come from.
People are weird / idiots.
In the case of Christianity the relation is a bit more complicated, of course, as Christian values coevolved with the same Western Culture(s) and values. Though in the late 20th and early 21st century these two are increasingly decorrelated. But I think it's not excessive to claim that overall, Christianity is the major religion that is closest to the modern definition of a 'good person' overall. Which is an idea worth considering, especially given that it would then follow that there would be a religion that is opposite to that. Food for thought.
But yeah, past that ultimately individuals are individual (to a varying degree) and except for some official decrees of religious leaders I don't think a good deed can be OF a religion, but rather strongly correlated to it. One exception I guess would be things done directly in the name of said religion, that if matching the precepts of said religion (holy books etc.) would have the fairest claim to be a deed 'of the religion'.
You may be oversimplifying the co-evolution of Christianity and western values. Christian institutions repeatedly fought against science, the enlightenment, egalitarianism, et al, then begrudgingly adapted and moved toward more moderate stances whenever it was clear they were losing the culture wars. Thereâ€™s always been a tension and conflict with the church first resisting and eventually coming around to evolving values such as democracy and adherence to the scientific method (and yes, I realize the scientific method was developed by a religious person, but overall it has never been of great importance to religious leaders and followers). I mean, for example, itâ€™s taken the Catholic Church how long to officially accept certain scientific truths about our planetâ€™s position in and relationship to the greater cosmos?
The USA is a more complicated case. Although the government itself was founded by men who were (mostly) influenced by the European enlightenment, the culture itself was greatly shaped and influenced by religious fundamentalists, many of whom fled from Europe as it grew increasingly liberal and â€œenlightenedâ€. The so called Great Awakening which occurred in colonial America during the mid 18th century also helped to shape the current culture. Itâ€™s shaped our culture to the point that even atheists on the political right here often agree with core Christian fundamentalist tenetsâ€”this is in contrast to many conservatives in nations like the UK
In short, I donâ€™t really consider the USA a western nation in the same way I might the UK, France, or even Canada. All of the other western nations essentially dumped their fundamentalist baggage and the most fervent believers and sects into the new world. Why a nation like Australia is a different case (and I realize Iâ€™m also oversimplifying and generalizing a bit now) is that they instead had convicts and poor farmers and ex sailers dumped on them rather than Bible thumping social conservatives