I assumed (perhaps incorrectly) that the OP only suggested substituting snipers for grunts with assault rifles. So you would still have anti-tank infantry, MANPAD infantry, etc.
I went back and reread the OP, and you're absolutely correct. In general terms then the situation is much less dire. Sniper rifles are still not really suitable for many situations, but this will be more of a handicap than a crippling blow to their ability to operate.
Moving to the psychology of it, the reality is that armies engaged in combat are already capable of killing quite large numbers of people, and army PR doesn't emphasise that. I'm not sure why the manner in which the deaths are achieved should change the way the role of the army is presented, unless that is changed as well. You might see something more like what happens in countries like New Zealand - advertisements for the army emphasise non-combat trades, comradeship, opportunities for travel, and allude to but do not necessarily spend much time on the combat arms. Still, most civilians are able to avoid thinking about the actual job of the army in any case, so I don't see that being too much of an issue.
The effects on the individuals are more problematic to determine. The army in general doesn't like psychopaths or thrill-killers, they're not reliable enough and don't work well with others (quite apart from the moral dimensions of the issue). But I've heard that snipers say that the face in their scopes isn't really a person to them, just an image that, with a few ounces of finger pressure, just... goes away. This implies a degree of emotional detachment from what they're doing, and if that same quality is required of all infantry it's going to differentiate soldiers significantly from the population they're drawn from.
Also bear in mind that
all soldiers are expected to be able to operate as infantry at need, and that means all of them will have had the same basic training in infantry techniques. This is the time when this sort of detachment would have to be fostered, so there might be implications for the other arms of service as well.