Here are the charts for 2019 and 2020 in the US*: https://imgur.com/a/HvBJoj2 (Note the axes are on different time scales unfortunately).
They are very similar with two major exceptions.
1. Week 13, about March 23rd, there is a dip lower, presumably due to the lock downs.
2. There was no activity throughout this spring and summer, but that's typically the case as the 2019 chart shows. 2020 is even lower (virtually no cases). While this could be due to lack of reporting, it could also be due to social distancing and masking.
We are currently in Week 44 of 2020. During week 44 of 2019 cases began to uptick slightly and then increased parabolically into the winter. We should expect the same in 2020. And this will be how we really know if flu testing is suppressed. Anyone claiming anything otherwise is speculating.
Anecdotally, my sister caught a bug recently that tested negative for COVID, and the physician suspected the flu. However, he couldn't test for flu since there was no capacity for flu testing.