Our old Victorian underground rainwater tank which held about 1000 gallons never smelled but the rainwater did go through a gravel and sand filter before the tank.

mines a bit pongy , but I haven't used it much

Depends a lot what's in it.
Which depends a lot on your roof, and the rate of rain.
Water from our greenhouse stays lovely and fresh; probably because it doesn't have much in it.
Similar storage of water from the house roof whiffs a bit - I would not drink it. But we have a 100+ year old clay tile roof covered in moss and lichen and bird poo.
Neither are filtered on the way in though. I expect that would make a difference too.

I expect a lovely new zinc roof would produce nice clean water. An old thatch, less so.

Is there any turnover? eg if you extracted at the same rate as rain was delivered it would probably stay fresher.

You might try fixing some copper wire along the ridge if you dont want the lichen growing.

We quite like lichen and are close to stream, so we used bronze nails for roof instead of copper to keep things tickety boo re wildlife.