I have to say I've no idea what I'd do if I couldn't send our child to daycare. I work from home, but past experience has taught me that I cannot work if he's home.
On the rare times he's been sick I stay home and take care of him, and don't even pretend to work. Too much "Daddy play with me", "Daddy look at this", etc. Even if he's basically sofa/bed-bound and sleeping all day I can't work.
At least in Finland we're allowed to have 9ish days of paid leave a year to take care of a sick child.
Most parents will say it gets loads easier after those first few years. Poor sleep makes everything harder.
It gets better.