A course of ABS is ten days normally. If your family has ear infections frequently then you are looking at weeks on end without getting out.
'Puking or shitting all over the place' is a bit of an exaggeration -- babies and toddlers wear nappies, which need changing. Small children who are just out of nappies sometimes have an accident. Small children sometimes develop a high temp or need some of their average ten day course of ABs administered in the evening before bed, which would happen if they were with a CM or in a nursery too, or with an au pair or nanny. I don't know anyone who can take ten days off just because their child has to take ABs, and that would be a ridiculous peoposition if your family was large and prone to ear infections.
Where babysitting is concerned, an ear infection that is being treated with ABs is very unlikely to cause any problems on day 4 or 5 or 6 of treatment and the DC who has it will probably be feeling quite well. Normal common or garden illnesses, the kind that afflict babies and small children regularly, are predictable and treatable and generally have no side effects, and with a competent babysitter who maintains order and is caring and kind, a child will be fine if parents are out for the evening.
Again, having a larger than average family can change your perspective, and give you an actual need to get out or it can all get very wearing. A refreshed mum who has had the chance to get out to her book group or a dad who gets out with his bowling team weekly will possibly have more energy to tackle family ups and downs for the next week. And the children themselves are often able to feel more secure in their large group than one or two children might be.
Apart from that one large family, my DDs generally babysit for families that have at most 3 children, and report that the smaller-family children are sometimes a bit more demanding/less well able to play together without squabbling, less well able to do things like bringing their plates to the dishwasher and co-operate with picking up toys, less aware of where things like more loo roll might be stored if they run out, or (memorably) where the plunger might be, and far more dependent on a long and complicated bedtime routine to get them to bed.
Flow, If you can afford to go out for a night and presumably spend a bit on yourself then you should be able to afford a fair amount for care for your children when you do so. Otherwise you need to rethink your budget for going out.
It's a pity to tar all teens with the same brush Nancy. You have encountered some spectacularly poor examples. 