A circulation pump for hot water. Yes or No?

No, the woman is wrong. The 300 are very well invested. Because then you have hot water immediately not only in the shower but also at all other tapping points.

Because you don't just shower but also need hot water for other activities. Letting the water run for about a minute until it gets warm each time is neither comfortable nor reasonable.

Small addition regarding usage: We also had a circulation pump in the last house. However, not with a timer. The electrician simply ran a few extra lines and installed switches in the kitchen and bathrooms. So just enter the bathroom, operate the switch, undress and jump into the shower and immediately draw warm water. I believe the circulation pump then switched off again after some time of non-use. I also found that a very charming solution.

We have none. The water is actually the same temperature. Although the bath upstairs has very short distances. The tablet would have to be kept in the shower.

The question arises whether it is a circulation pipe for 300€ or some kind of heating cable that keeps the pipe warm from the outside, because such things also exist!

I would save the money for "smart" pumps or switches. It's not worth it. Such a pump needs a maximum of 10W. Even in continuous operation, that's only 25€ of electricity per year. Absolutely negligible.

Our circulation pump is connected to the heat pump and can be time-controlled through it. It is off from 10 PM to 6 AM. Otherwise, it runs continuously.

I would save my money on "intelligent" pumps or switches. It's not worth it. Such a pump needs max. 10W. Even in continuous operation, that's only 25€ electricity per year. Absolutely negligible.

Our circulation pump is connected to the heat pump and can be time-controlled via it. It is off from 10 p.m. to 6 a.m. Otherwise, it runs continuously.

It's not about the pump's electricity, but about the heat loss from the circulation line!