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I used to have a tankless water heater and loved that I never ran out of hot water in the shower. What I didn't like is that it took about 60 seconds for the water to get hot, which is really annoying when I just wanted to wash my hands with warm water.
Is there a system that combines unlimited and instant hot water? Maybe a hybrid – an on demand unit with a small, heated tank?
The tank is not the issue. The distance and pipe size is the issue.
There is a considerable length of "hot water" pipe between your hot water heater and your shower or sink. That pipe is always full of water. Over time, this water cools off. It is now cold water, even though it's in the hot-water pipe.
When you turn on hot water, hot water starts flowing into the pipe from the far end. The cold water already in the pipe must be pushed out before you feel warm water at the faucet. How long depends on faucet flow rate, length of pipe, and pipe diameter squared.
That "square" rule is a real problem. Go from 3/8" to 1/2" pipe, your delay doubles. Go from 1/2 to 3/4, it doubles again. And some dummies think 3/4" pipe is deluxe and will give them better showers. Nope, it just makes them wait. Downsizing pipe as much as possible will help.
Also, you can move the hot water heater closer to (right next to) the shower or sink. That's easy with electric tankless heaters. However in practice, most tankless heaters are simply installed where the old tanked heater had been (which itself was chosen because of the ugly, awkward bulk of the unit) -- in which case obviously there's no speed improvement, and speed may be worse since flow rate is limited on on-demand units. The solution is to install the tankless unit much closer to the point of use.
Tankless water heaters are great unless you have a large home then multiple units are needed. In my early years we built very large custom homes and installed recirculating pumps that cycle the water in a loop using the cold water line the only problem here is that the cold water is now warm and it takes more power over all but the units are not that expensive and quite easy to install on the furthest hot water tap from the water heater here is an example. If your home has 3/4" pipe to prevent the pressure drop that causes a shower to go cold when some one uses water in another part of the home this is one way to not have to wait for minutes for hot water. The timer will need to be adjusted based on the length and size of the pipes and if the pipes are exposed under the home they should be insulated to reduce the power consumption but this is one way to have hot water quickly. The only drawback is I don't think they will work with a tankless system but not positive about that.
As @Tyson said, that's water in the pipes that had cooled down to ambient temperature. The water coming from the heater is hot, it just has to get to the shower, flushing the cool water out in the process. There's not a whole lot you can do about that without moving the heater closer to the shower.
There are on-demand inline water heaters that install under your sink, and provide instant hot water to that faucet only. Since it is so close to the faucet, there is little cool water to flush out of the pipes until the hot water gets there.
There may be similar units you can install for showers. You'd probably need an access panel of some sorts, either on the shower wall, or on the opposite side of that wall.
