That's been tried.
There was a helipad on a barge near the City about 12-15 years ago. If I remember correctly, the theory was that, by periodically changing position, no planning permission would be required. Predictably, the noise lobby came out in force and it was stopped from operating.
You're right that it shouldn't upset too many people but, unfortunately, it does. The number of people living along the River and affected by helicopter noise is relatively small, but that doesn't stop noise protestors who live more than a mile away influencing the local authorities' decisions. My own borough, Kensington & Chelsea, is a classic example.
The three relevant local authorities on the North side are K&C, Westminster and the City of London - each of which has some influential people either living or working locally. I doubt if either of the two main political parties would help - supporting helicopter ops isn't exactly a vote winner.
If Battersea closes, I suspect the chances of another heliport in or near Central London are pretty slim.
(Edit)
I also have a vague recollection of reading that there was a pontoon on the Thames (near Lambeth Bridge?) in the 1950's. I don't know what happened to that.