That would be best, but a tankless heater needs a lot of energy to heat the water on the fly so would need a high ampacity electric line or a gas line (plus exhaust). The little 2.5 gallon heater I use share a circuit with the garbage disposal. By the time the 2.5 gallons is used up the water from the main heater has arrived.
Where I am, cold weather isn’t a problem. It is reaching 45 C today. 
But I did take the time to wrap the outlet of the HWS for a goodly length. HWS is a heat pump driven storage system. COP is 6, which is pretty good. Gets run off solar power in the middle of the day. So hot water is nearly free most of the year.
What I had noticed was that the outlet copper pipe was aways quite warm some distance from the storage tank. Clearly there is a constant heat loss occurring from the tank. Not huge, but not insignificant. So the OCD in me wrapped it up in pipe insulation along the length before it enters the house. I don’t think it makes any noticeable difference to the time it takes for properly hot water to arrive when needed, although in principle it might clip off a second or two. None of the runs are are particularly long, so it never bothers me. All of the pipes are copper, and none are insulated inside the house, so the pipes provide no help keeping temperature.
I did not find any material increase in gas costs running my recirc system. I was then living in a 4-seasons area where my HVAC costs were $300 / month pretty much year round for heat, cool, or many days both. Plus the year round base load for lighting, appliances, etc.
I do recall the tepid cold water in the kitchen to be surprising at first and occasionally annoying. But not something I thought needed an additional workaround. YMMV of course.
My house had the HWH in the unfinished part of the basement which ran the breadth, but not depth, of the house. But run under all the places where HW (& CW) risers went up to the floors above. I was able to easily insulate the exposed HWH pipes running along the basement ceiling for the ~100 foot run to the far end of the house. I suspect that mitigated a bunch of the incremental gas cost.
I suspect you didn’t often drink directly from the cold water tap, either. 
Not in the kitchen since the advent of a cold water dispenser in the fridge door. But I did & do plenty of drinking from the cold tap in the bathroom at night or first thing in the morning or while getting cleaned up. 
Having grown up in rather warm SoCal, I found moving to the 4-seasons Midwest a strange experience. Getting actually cold ~40F, not ~65F, water from the “cold” water tap in winter was a very mixed blessing, but net negative. Nice for a quick drink; sucked to wash hands. Having fast hot water was especially welcome when the cold side was actually cold, not my desired room temp.
My bedroom is a long way from the water intake. If i want cold (not room temp) water from the cold tap, i need to wait nearly as long as to get hot water from the hot tap, which is to say, i pretty much always wash my hands in tepid water, but often brush my teeth with cold water.
Also, I’m happy drinking room temp water (but not tepid water) and neither of us puts ice in drinks, so we choose not to hook up the fridge to the water supply, and removed the ice-making equipment to give up more space in the freezer. We drink from the kitchen tap all day long.
I also cook with hot water from the tap. But EPA and CDC warn against lead leaching into hot water from pipes. Pipes are legally allowed to contain up to 0.25% lead. I don’t know in reality how much lead modern pipes really have. There is also some concern about bacterial growth in the water heater tank. Again, I don’t know what reality is, or whether this is all pearl clutching. I have seen a post or two elsewhere by pros who work on these tanks who also recommend not drinking it.
We only use gas for water heating (by far the biggest part of total usage), clothes drying, and cooktop. The increase was obvious.
Well, my pipes are mostly ~70 years old, and my municipal water company just warned me against drinking the hot water. I’m going to continue to avoid it. I also run the cold water until I get past the room temp water to fresh water before filling a pitcher, and usually for other drinking/cooking purposes.
But even if I had all new plumbing, hot water tanks die because they grow crud on the inside, mostly. I just don’t want to drink that.
That crud is limescale, and you’re drinking less of it if it’s condensing on the innards of the water heater. Your cold water still contains the full complement of your local mineral content.
Lime scale, and also rust and random other stuff that precipitates out of the pipes and the water. If i cooked the dirt in my backyard in my oven for half an hour, it would be perfectly safe to eat, too. (I tested for lead before moving in.) It’s still not very appealing.
You’re missing the point.
Hot tap water contains less of the stuff you’re worried about, because much of it is precipitating in the tank.
The tank itself isn’t creating new stuff. It was in the water supply from the beginning, and the tank is actually removing some of it.
I’m not missing the point. First, that’s not entirely true, some of the crud is actually the tank, corroding into the water. Second, the hot water tastes nasty. Much moreso than the cold water, even after it sits and warms to room temp, which it typically does in my pitcher, which lives on the kitchen table. And third, hot water does pick up more lead than cold water, and my pipes are old enough that that’s a potential issue.
Every so often (year or two) the municipality would flush the hydrants here to get rid of the ultra-fine precipitate in the mains. Then, for the next houror so after they did your area, your water would run brown. So some miniscule amount of that is always in your water, espcially busy times when everyone is using their water. Plus red deposit indicating rust…
Out of curiosity, how much was the difference? I have a recirc system and I love it. Nearly instant hot water in the whole house. When I got it, I had gas heating so in the cold months the gas bill was overwhelmed with that. Now I have an electric heat pump for heating so I am like you with just water heater, range and clothes dryer. My gas bill is very cheap now. Clearly it’s using some extra gas but my total bill is $38/month.
I just use it for…cooking.
It’s been a few years, but I recall going from ~$40 to ~$80, then back down when I restricted the recirculation to 3 hours/day. Our water pipes mostly run below our slab, so might cool down faster than pipes run through a basement.
Thanks. I can’t access my bills from a few years ago to see if there is a significant difference in therms from before during times when I was not using the heater. I have a raised foundation house in a temperate area which may make a big difference. Also, back to the OP the plumber put pool noodles around the pipes for insulation.
My house was built in 1960 but I have replaced all of the plumbing in the last few years. The incoming is all copper and they stopped using lead solder in the mid 80s. Santa Barbara tap water tastes horrendous so anything I drink or use in cooking is filtered.
Pretty much for sure on that. Pipes buried in dirt lose heat a lot faster than pipes running in a heated basement. Of course the colder your outdoors environment the worse that is.
I just had the old lead water service lines in two houses replaced under a grant program administered by the local Water Supply Board - a great deal and I really appreciate it. Since then I’ve gotten repeated mailings that “they have no records of what the pipe that runs into my house is, and it might be lead” for those two houses, as well as matching letters for a half dozen others I paid to have replaced 20 years ago. I sent them scans of signed-off approvals from their own inspectors, but those apparently are not part of their records, nor considered adequate proof. I invited them to come check, and they wanted to make separate ‘morning’ or ‘afternoon’ appointments for each of the 8 buildings, which are all on the same street…
In order to stay on topic - yes, I insulated all the new plumbing that I put into each of these gut-renovated buildings, but mostly because I have OCD and was doing the work myself. Insulating pipes well is labor intensive, but the materials are fairly cheap.