When the BER = \$10^0\$ this means for an average number of bits, not each bit.
If you have random data BER will always be 0.5 and only message error rates can approach MER=1.
BER=1 is just an asymptotic value. (How about 1.1? J/K )
Thus inverting a random number of bits yields the same result. Message error =1.
However, you are getting close to understanding this concept that the other answer ignores.
When looking at different BER curves for the same system but due to different effects such you may notice a sharp difference near BER = 1 to BER = 1%. This depends on, for example, Trellis coding or RLL encoding or randomization or discriminator asymmetry or data pattern (worst case vs best case vs random) or some other eye pattern affecting the property is inherent in the de/modulation scheme.
In the "old days when data was encoded by MFM for magnetic storage without RLL encoding where the edges between or synchronous to clock edges determined the data value. Thus the data pattern had a strong impact on BER for the same SNR. This could easily be measure in very few bits or near BER= 1e0 at some low SNR. At this low SNR level, the jitter quality of the clock recovery also added to the noise of the data significantly in some designs more than others.
Thus for linear Bi-phase or NRZ or RX or HDD MFM modulation schemes, the worst-case pattern was always "6DB" hex or in octal 011 011 011 which put most of the change in frequency where the group delay changes most rapidly in non-ideal channels from 1f to 2f or 00110011 to 1010101 in half bits or full bits. There were other worst-case patterns too.
Although not done at BER=1 bur more like BER=0.1 and the slope of BER/dB or dB/BER changes with all the above variables. So discriminator asymmetry would easily show up for the difference between random and 101010 data by a few % of the bit in phase margin or ratio of bitshift/period of a bit Thus ALL GOOD communication channel designs must have a phase margin budget for each of the constant variables that degrade the channel other than Gaussian noise. This applies to both the data and the clock channel combined.
When you learn to find the weakness in any design, you will often do so by measuring the Window Margin aka phase margin or BER shift or slope shift due to some parameter change.
This was my strength in reverse engineering complex communication channels during Design Validation or Verification Testing formerly called DVT to see if the design margin to each design parameter would stand up against each stress factor from the environment (climatic, mechanical, electrical) to the channel medium fading or ageing or adjacent channel interference.
Thus it is not the certainty of inverting each bit in error when BER=1 that is wrong but it is wise to view the shift in BER near BER = 1 to 0.1 due to the aforementioned variables and examine the change in slope other than simply random data under benign conditions. Often did a quick test with 4 corner environment and voltage margins with vibration, and RATHER than simply test if it worked pass/fail, I would measure the comm channel Window margin so see how much margin was lost for each variable.