I would argue that you're not saving energy either.

The tankless itself isn't so much the cause of the delay as is the simple distance from water heater to the point of hot water use. A traditional tank unit would also involve a delay due to the distance. Bad plumbing design moreso than fault of the water heater. Even so, unless your house is exceptionally huge, I'm assuming 10 minutes for hot water to arrive is an exaggeration. If that's true, then you should for sure have a separate water heater for the bathrooms.

There are retrofit recirculation pumps that install under a sink for example, and feed from the hot line into the cold for a closed circulation loop so no water is "used" They're typically run either on-demand (a few mins before hot water is needed), or set on a timer to run every 15 mins or so, just long enough to purge the line. A fairly large-capacity pump may be needed for a long distance run to generate enough flow-rate to activate a (gas, I assume) tankless.

Echo what dadoes said:

This is bad plumbing design - not a bad water heater.

Tankless water heaters add about 4 seconds to the wait for hot water. It takes two seconds for them to put out heat and another two seconds for them to get hot water into the first 10 feet of pipe or so that would have been filled with hot water if it had been a tank system.

Yes, insulating your pipes will help reduce the heat loss between uses and keep the hot water in the pipes.

If your house is really large, then you should consider a circ pump (fixes the symptoms of bad plumbing design) or a separate heater.

Interesting. Hmm. There is no real wait for the hot water in the kitchen, just in the bathrooms at the other end of the 2300 square foot house. Perhaps, then, we really should install a hot water heater on the far end of the house for the bathrooms. We used to have two huge hot water heaters, tanks. We never lacked for hot water. No joke about the 10 minute delay. Before showering, we literally have to turn on the shower, go find something else to do, and then return. The water slowly warms up. It goes from bone chilling cold to tepid to finally hot.

I imagine we could build a cabinet for a tankless water heater on the outside of the house on the affected end of the house? Our basement is only under the kitchen. That is where we typically locate water heaters, but, now, that seems ineffective for the bathrooms.

Is it a fallacy that tankless systems save energy?

Gas tankless units save about 38% over a standard gas tank for the same amount of water.

Electric tankless units only save about 3%.

The gas tankless save, not so much because of the lack of a tank, as much as the lower flue temps. Not having an uninsulated chimney up the center of the tank helps also.

A standard "energy saving" tank unit has a energy factor of .62 a regular, non-condensing tankless unit has an energy factor of .84 -.86

Do you have uninsulated pipes in a slab? That would be a serious problem.

They can save energy if used carefully/properly/intelligently, but most people don't, or won't, change their hot water usage habits accordingly.

My house is 2,550 sq ft. An L-shape layout. My master bath is at the far end of the long leg. Kitchen at the bend of the L, with my electric tankless being in the laundry room at the near end of the short leg. 2nd bath is at the far end of the short leg. Master bath is longest to get water, of course. I generally don't use the master bath except occasionally in summer, and I'd never formally timed it. Checked just now. It took ~8 mins to get warmed water at one of the master bath sinks, although that was at only 0.5 GPM flow rate. The washing machine is much quicker ... so yeah, it's all about the distance.

Note: My house was already built when I bought it, I didn't have any input on the design.

iF YOUR PLUMBER followed goverment regulations, your shower faucet will have a stop which limits the max temp of your shower. It does this by limiting the percentage of hot water that comes out when the faucet is turned all the way to hot. What that percentage is depends on the temp of your hot water AT THE TIME THE PLUMBER IS DOING THE ADJUSTING. If the temp is very high, you would get a very small volume of water flowing thru the hot water pipes when you have the faucet wide open and turned all the way toward hot. You would at the same time get a relatively large volume of water flowing thru the cold water pipe.
This can be "fixed" by removing the temperature stop from your faucet.

I have a 2400 sq ft single story house with the shower at the oposite end from the water heater and low water pressure. I get hot water in about 2-3 min. If your's takes 10 min you have other problems besides poor plumbing design.

Another thing to check: Some single handle faucets will allow cross flow between hot and cold. When you turn on the faucet in your shower, hot water pressure drops more than cold due to restrictions thru the water heater and extra corrosion in the pipes. This can cause cold water to flow into your hot water pipes thru the faucet or shower in your other bath or kitchen. This is usualy easy to check and sometimes fix. Just turn the unused faucets all the way to hot or cold. If this is not possible, close one of the shut off valves under the unused faucet.