depending from the POV.
if a condition has an incidence of 0.001%, production capacity of test kits will probably be geared to that.
if you suddenly have 0.01% false positives, you're outstripping production by a factor of ten
it will also use up production capacity that might be necessary for other kits, driving up both test prices and/or wait time for everyone else.
for example: false positive in Italy are the reason you see a new case slowing down in the last week: https://www.google.com/search?client=firefox-b-d&q=italy+cov...
this is not because the containment measures are working, it's because a lot of tests are redirected to verify the PCR results, eating up test capacity and slowing down for example contact tracing tests, which are competing for the same kits.
so, no. false positives have far more implication than one individual short time of anxiety.