Using GNU Fortran 10.3 with the standard intrinsics, and asking the Fortran runtime library to peek the seed, it seems that every invocation of the program does result in a different serie of random numbers. So that would be OK for the sort of application you have in mind.
Using this code:
Program TestRandom1
  implicit none
  integer :: randomSeedSize
  integer :: count = 3
  integer :: k     = 0
  real    :: rx    = 0.0
  call random_seed(size = randomSeedSize)
  write (*,'(a,i4)')   'size of random seed (in integers): ',  &
                       randomSeedSize
  call random_seed()  ! use system-provided seed
  do k = 1, count
      call random_number(rx)
      write (*, '(a,f10.8)') 'rx = ', rx
  end do
End Program TestRandom1
Context:
$ 
$ uname -s -m -r
Linux 5.13.9-100.fc33.x86_64 x86_64
$ 
$ gfortran --version
GNU Fortran (GCC) 10.3.1 20210422 (Red Hat 10.3.1-1)
Copyright (C) 2020 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
$ 
Testing:
$ 
$ random1.x
size of random seed (in integers):    8
rx = 0.23642105
rx = 0.39820033
rx = 0.62709534
$ 
$ random1.x
size of random seed (in integers):    8
rx = 0.84118658
rx = 0.45977014
rx = 0.09513164
$ 
$ random1.x
size of random seed (in integers):    8
rx = 0.33584720
rx = 0.86550051
rx = 0.26546007
$ 
A seed size of 8*32 = 256 bits looks consistent with the xoshiro256 algorithm mentioned here in the GNU Fortran documentation.