Both sound bad to me, but I can to a certain extent understand the logic behind the first option in terms of the financial interests of the company. I can definitely see a "Sorry Ashley, you're out of paid sick days so we won't be paying your salary this month, we can't afford paying for nothing. I hope you recover and can come back to work with us soon!". The way we make up for this in Spain (and I'm assuming in varying similar ways in other countries) is by having the Social Security start taking charge of growing percentages of the employee's salary during sick leave as it prolongs in time, so as to reduce the impact on the business.
The second "drop you" sounds much worse though. You have an illness or an accident bad enough that it exhausts your paid sick days, so your employer simply sacks you out of impatience? And what happens to the employment-dependent health insurance which (I assume) was what was covering you on your long sick leave? "Sorry Ashley, you're out of paid sick days so we're firing you. I hope you recover enough from your severe TBE before your savings run out so you can find a new job. Also I hope you weren't depending too much on our health insurance plan for rehab. Good luck!".
When this happens you lose your insurance too, usually at the end of the month. You can pay around $800-1200 a month to keep it though, more if there’s a spouse or kids