My understanding is that the older, smaller varieties are better tasting than the hybrids that go for size over everything else.

The strawberries I grew were from a plant exchange and they were an heirloom, smallish variety. I was a failure at keeping the patch up (I take all the blame), but in the two years I did get some berries, they were very flavorful and of the deepest, darkest red I have ever seen on a strawberry. Maybe you could get another variety to add to your patch in the Exchange forum. Then you could judge for yourself if variety is the problem. Carmellia

What affects the taste? Soil? Weather? Variety?

All of the above. And especially plant health. Struggling plants will not be flavorful, or very sweet. Strawberries planted incorrectly, i.e. too deep, would be struggling. This is probably the most common mistake made.

It is known that with strawberries in general, more acidic soils will produce sweeter berries.