Yes, but some smaller businesses might not be able to afford it. So put a number on the number of employees a company must have before it is mandatory (I'm not going to).
The problem with that is if small businesses won't offer paid sick time as a benefit, they may struggle to compete with the bigger companies in the hiring market. Rather than be exempted from a federal law mandating paid leave, the government would step in to cover the tab for businesses under a certain size.
Your first sentence is absolutely correct. But that's always been the case with small businesses trying to compete for talent. They don't have the money the big boys due for great pay and benefits. However the smart small businesses will, if they can afford it, offer it. For my part time assistant, I pay her when she's sick, and she's making much much more than her past job. There are other perks she has too. But it's my choice to do it and she's outstanding so I do what I need to not to lose her.
I'm just not in favor of a government mandate for small businesses (pick an employee count number to define it) or another government program, to be honest. So I support it but I'm just not sure if it's doable.