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It's not really a question of economics or environment (well - the latter maybe slightly)
It's more to do with the fact that you get 45 seconds of cold water that's been sitting in the pipe between the boiler and your tap, before the heated water arrives where you want it. I get the same in my place - having to wait even longer for hot water in my bathroom as that is even further away from the boiler. Those under sink water dispensers produce water that is way too hot for the needs you describe. Also, if you are in a hard water area, they get furred up with limescale a treat.
Where the process is wasteful is when you have finished washing your hands or whatever, the hot water that your boiler has worked hard to heat which is sitting in the pipe between the boiler and the tap, now cools down until it once again becomes cold. You could mitigate this to a small degree by insulating the pipe run between the boiler and the tap. -
You could also use an insulated kettle for hot water. I just bought this Vektra one: https://vektra.uk/collections/series-1/products/vektra-1-5l-stainless-steel_11
Too early to say how well it copes with SE limescale, but it certainly does what it claims in terms of keeping water hot for a couple of hours.
We have to run our kitchen hot tap (and therefore boiler) for about 45 seconds before the water gets warm. What’s better economically and environmentally; this, or one of those under-sink boilers that dispenses instant hot water? Just for handwash and washing-up.