I'm first not sure of the question's premise; I am unable to find the quote in any of the scholarly sources I have, and I also combed primary sources from 1894-1896. I've only able to find it in pop online web sites, like pankration.net which dates the quote to 1895 and gives it in French as "Nous acceptons tout, sauf le pancrace".
The thing that makes the quote doubly odd is a.) a draft list of sports was already made by then b.) it never included pankration in any version and c.) it isn't true that all ancient sports were included. Boxing didn't make it in until a later year (more on that in a second) and unless you've been living in a parallel reality, the modern Olympics has never had chariot racing.
So I'm skeptical, but I'm also not entirely convinced the quote was false, as I am missing access to a number of secondary sources in French. However, what I'm guessing you really want to know is:
Why did pankration not make it into the Olympics?
The "father" of the modern Olympics, Pierre de Coubertin, had been pushing the idea since 1892 (with a speech at the Union des sociétés françaises de sports athlétiques) but movement really started in 1894, and the first time we see a list of proposed sports is in the Bulletin du Comité International Des Jeux Olympiques -- a bulletin from the Olympic committee -- circa July 1894. Translated:
The following sports ought to be represented, if possible: Athletic sports (track & field), Aquatic Sports (rowing, sailing, and swimming), Athletic Games (football, lawn tennis, paume, etc.), Skating, Fencing, Boxing, Wrestling, Equestrian sports, Polo, Shooting, Gymnastics, and Cycling.
There should also be instituted a general athletic championship under the title of pentathlon.
At the occasion of the Olympic Games, an alpinism prize should be awarded to the most interesting climb accomplished since the last edition of the Games.
Notice that boxing and wrestling are included, but not pankration. Between this point and when the actual event was held, the list had some instability (Cricket and Polo were planned, but somehow never materialized) and even during the Olympics there were changes; rowing was planned but canceled due to bad weather, and of the fencing disciplines, while saber and foil happened, épée didn't occur because foil ran for too long.
In other words, the list of events was a bit disorganized. As far as why boxing was left off, Pierre de Coubertin wrote shortly after the Athens games happened that they
... substituted bicycle for chariot races, and fencing for the brutalities of pugilism, but because in their origin and regulations they were international and universal, and consequently adapted to the conditions in which athletics have developed at the present day.
Again, not even mentioning pankration, although another writer from that year (Richardson in the September issue of Scribner's Magazine) noted
An ancient Greek, had he come to life again, would have missed some of the events of his old games. The pancration, with its brutalities, was happily lacking. Even boxing was omitted.
So: boxing was briefly considered too "brutal" a sport to be appropriate for the Olympics. Pankration, as a combat sport with the only rules being "no eye gouging, no biting", where there was no time limit and only the opponent's forfeit, knockout or death ended the match: the elements would definitely be considered a bridge too far.
The other major obstacle would have been finding any competitors: it wasn't a living sport that people played at the time. There wasn't any modern version until the athlete Jim Arvanitis popularized it starting in the 1970s, and while there are now actual rules developed, it wasn't the sort of thing doable on the spot in 1896. (Rulesets can now vary, but added to the eye gouging and biting are prohibitions like "hair pulling", "spine twisters", and "small joint manipulation".)
In summary:
1.) nobody ever proposed having the sport at the time
2.) even boxing was originally considered too brutal to be in the first Olympics
3.) nobody played the sport in 1896
4.) nobody would have known what ruleset to use
...
Mallon, B., & Widlund, T. (2015). The 1896 Olympic Games: results for all competitors in all events, with commentary (Vol. 1). McFarland.
You can read the various bulletins of the Olympic committee at the time at the Olympic World Library.