Rich kids don't feel compelled to enlist just to make financial ends meet. Even I (single parent household, parent was a teacher), when faced with an inability to find a job when out of high school, had the grades and support to 'fall back' onto college instead. I wasn't forced to risk my life just to stay solvent.
More on point, though, I think a lot of people with this view conflate the particular implementation of welfare (itself a cause of compromise between right and left wing), with the nature of welfare itself. That is, because there is such a wide gap between "welfare, not working", and "fully self supported" that unless you can land in a job that pays well it doesn't make sense from a total utility perspective for many to find a low wage job and work that in lieu of (or alongside partial) welfare, we dismiss the idea of welfare completely. Or by example, the person on disability who is unable to work full time, but who, if they try to work above X amount of hours, ends up -losing- money, due to loss of disability benefits, as well as (obviously) losing on free time, and so ends up not working as much as they could. Due to cases like that, many people end up blaming welfare itself, rather than a flawed implementation of it.
Per another commenter, I too think a UBI would address this (as well as the basic issue that with more automation we have less need of people working, and I'd prefer someone trying to figure out what useful work they could do that would result in them getting paid for the real utility they provide, rather than just taking an unnecessary job just to make ends meet. Wage slavery in jobs that could be automated is an even more flawed form of welfare than government handouts).