Is it really going to damage a bike to wash it with a hose?
Yes, pressurized water can damage your bicycle, specifically by washing the grease out of bearing areas. Note, this is pressurized water.

Using a garden hose without a sprayer is unlikely to do damage. Using a garden hose with a shower type sprayer is unlikely to cause damage. Low pressure, so that it doesn't break the seals, is the point. (I avoid the risk, anyway, because I can't afford to be wrong, and how much pressure is OK depends on the strength of the seals on your particular bike.)

A car wash, or any other pressure washer, is pretty much guaranteed to do damage.

Washing by hand, with a bucket, a car wash brush, a toothbrush or gear brush, and some non-detergent based soap is the safest way, and with a little practice and preparation, can be done in half an hour.

Some detergent based soaps react to aluminum, and can cause serious damage to an aluminum frame or parts over time. Make sure you use an aluminum safe soap.

You can make things easier by using a repair stand to hold the bike, and removing some parts, if you have the tools and knowledge, but that's a bonus.

Other than that, it's just elbow grease.