Strange result with Laplace transform

The following code:

f[x_] := HeavisideTheta[1 - x]/x
InverseLaplaceTransform[ f[t], t, x]/x

Returns 1/x. But this should not be the case because the function 1/x is invariant under this transform and its inverse LaplaceTransform[ t f[t], t, x]. Applying the later transform to the result should return the original function.

Is this a bug?

UPDATE

With GenerateConditions, the operator InverseLaplaceTransform[(1-E^-x)/x, x, t, GenerateConditions -> True] gives 0 which is wrong (without options the result is correct).

UPDATE2

Tested on Wolfram Engine 12.1, it seems to give a correct result, so this is likely a 12.1->12.2 regression.