did the invention of the Fischer mode (meaning you must then notate even in time trouble) change anything?
Most certainly. It meant the virtual demise of the infamous "10.2". This in turn meant that the job of the arbiter became much easier and the game became much fairer.
"10.2" does live on in the FIDE Laws of Chess but it has moved from the "Laws" to the "Guidelines", specifically "Guidelines III. Games without increment including Quickplay Finishes" and it is likely to disappear altogether in the next version.
First what is a "quickplay finish"?
III.1 A ‘quickplay finish’ is the phase of a game when all the remaining moves must be completed in a finite time.
What are the unjust rules in this case?
III.4 If the player having the move has less than two minutes left on his clock, he may request that an increment extra five seconds be introduced for both players. This constitutes the offer of a draw. If the offer refused, and the arbiter agrees to the request, the clocks shall then be set with the extra time; the opponent shall be awarded two extra minutes and the game shall continue.
III.5 If Article III.4 does not apply and the player having the move has less than two minutes left on his clock, he may claim a draw before his flag falls. He shall summon the arbiter and may stop the chessclock (see Article 6.12.2). He may claim on the basis that his opponent cannot win by normal means, and/or that his opponent has been making no effort to win by normal means:
III.5.1 If the arbiter agrees that the opponent cannot win by normal means, or that the opponent has been making no effort to win the game by normal means, he shall declare the game drawn. Otherwise he shall postpone his decision or reject the claim.
So, in this case it is not the play of the players which decides the game but the decision of the arbiter, who is often a much weaker player than either of the two players, who decides the result of the game. Note that neither player has broken any rule nevertheless the arbiter can bring the game to a premature close on the grounds that the game cannot be won by "normal" means.
By extending the game time allotted by 30 seconds every time a player makes a move (Fischer mode) the game becomes much fairer because the players decide the game not the arbiter.