I guess I would start by saying I reject your premise. Though there are undoubtedly both conservatives and progressives who share the view of humanity you ascribe to them, in neither case does such a view necessarily follow from their respective views on the role of government.
Since I consider myself to be more on the progressive side of the spectrum than the conservative one, I will start there. My own view that government/society should provide assistance in particular instances isn’t motivated by a belief that poor people somehow can’t be trusted to do the right thing with the resources available to them. For more on my views on poverty I refer you to my Medium piece https://firstname.lastname@example.org/who-are-the-undeserving-poor-when-i-meet-one-ill-let-you-know-30cd7a02bb4.
Millions of children are born into poverty each year, for example. Even if we accept the argument that these children’s parents make terrible decisions on a regular basis, the children themselves must be blameless. Starting off life in a home where food is scarce or educational opportunities few and far between puts even the most naturally gifted children behind the eight ball. Outside intervention of some sort is therefore necessary to level the playing field to a sufficient degree to give many of these children a reasonable chance in life.
As for conservatives, I regularly hear the argument from their own mouths that they believe humans to be largely motivated by a kind of crass self-interest. It is precisely because government is run by people that they distrust government. They do, I think, tend to regard individuals as better informed or better able to decide for themselves what is best for them than a government agency or bureaucrat. I agree on that front. Again, see my article on the so-called “undeserving poor” cited above. That said, conservatives also frequently argue that regulations on how Food Stamps or other assistance is used are necessary. They are, after all, very fond of pointing to instances of individuals buying junk food or lobster with government money to argue on behalf of greater policing of the poor. Welfare Reform in the 90s was, in large part, a response to Reagan’s mythical “welfare queen.”
Regardless, as I’ve pointed out in an earlier response, self-interest and cooperation are hardly mutually exclusive. It’s in society’s self-interest as well as our own individual self-interest to maximize access to healthcare, education, nutrition, etc.