Without acknowledging any of the conditions actually present in the church, wood fires can get much hotter than 600 °C.
The maximum temperatures measured within the pile were of the order of 800, 1000, and 1200 °C for piles composed of 1.27, 2.54, and 9.15 cm sticks respectively, although the maximum temperatures for a given size stick appeared, from all data obtained, to be somewhat dependent upon the structure of the pile. The prescribed temperature-time curve of a standard fire exposure test 1 is also shown in figure 4 from which a general agreement may be noted.
D Gross: "Experiments on the Burning of Cross Piles of Wood", Journal of Research of the National Bureau of Standards- C. Engineering and Instrumentation Vol. 66C, No.2, April-June 1962. (PDF)
A nice pile of wood with good ventilation can get apparently really hot:
Fire plume temperature data suggest a maximum turbulent flame temperature in fully developed compartment fires of about 1500 C for stoichiometric and adiabatic conditions. Experimental results for crib and pool fires are presented to support the trends indicated by the approximate analyses.
In general, from equations 12 and 13 for stoichiometric conditions, the temperature is given as
where Tf,ad is the stoichiometric adiabatic flame temperature. Recorded gas temperatures near the ceiling are reported as high as 1350 C , and mean temperatures over the peak burning period are 1000 to 1200 C for polyethylene fires  and approximately 900 to 1200 C for wood cribs . For turbulent fire plumes, having a radiative loss fraction Xr, a similar formula applies to the combustion region. This turbulent flame (centerline) temperature is given as 
From the best available data [22–24], the turbulent mixing parameter, kT, is found to be approximately 0.5 for cp 1 kJ/kg K. As the fire diameter increases, the radiative fraction falls due to soot blockage . Fig. 4 shows flame temperature data for turbulent plumes as a function of Xr. The extrapolated adiabatic temperature is approximately 1500 C. For a realistic adiabatic flame temperature of 2000 C, the actual turbulent mixing factor is approximately 0.75 or a turbulent dilution factor of 1.5. For a large fire in a compartment with large vents, the core maximum flame temperature should approach the turbulent adiabatic flame temperature.
James G. Quintiere: "Fire Behavior in Building Compartments", Proceedings of the Combustion Institute, Volume 29, 2002/pp. 181–193. DOI
But to be very sticklish, the claim is actually somewhat correct. Why?
M. J. Spearpoint And J. G. Quintiere: "Predicting the Burning of Wood Using an Integral Model", Combustion And Flame, 123:308–324 (2000). DOI
Or to put it simply:
A bonfire can reach temperatures as hot as 1,100 degrees Celsius (2,012 degrees Fahrenheit), which is hot enough to melt some metals.
Most types of wood will start combusting at about 300 degrees Celsius. The gases burn and increase the temperature of the wood to about 600 degrees Celsius (1,112 degrees Fahrenheit). When the wood has released all its gases, it leaves charcoal and ashes. Charcoal burns at temperatures exceeding 1,100 degrees Celsius (2,012 degrees Fahrenheit).
Gabriella Munoz: "How Hot Is a Bonfire?", Sciencing, April 26, 2018.
This is a rough guide to flame temperatures for various common substances (in 20 °C air at 1 atm. pressure):
Wood 1,027 °C (1880.6 °F)
Methanol 1,200 °C (2192 °F)
Charcoal (forced draft) 1,390 °C (2534 °F)
and gives for adiabatic flame temperature maximum even:
Wood Air 1980°C 3596°F
The French authorities and experts seem to have suggested that inside the church, around the altar, theoretically 800 °C might have been reached, with 'improper' firefighting methods, meaning that it stayed cooler in that area. Or that even temperatures of 2000—2500 °C occurred actually in the roof where the main part of fire burned:
Contrairement aux pompiers américains, les sapeurs-pompiers français s’attaquent aux incendies par l’intérieur et non de l’extérieur. Cette tactique est plus dangereuse pour les hommes mais plus efficace pour sauver le patrimoine, observe l’expert Serge Delhaye. Si l’on se concentre sur l’extérieur, on prend le risque de repousser les flammes et les gaz chauds, qui peuvent atteindre 800 degrés, vers l’intérieur et accroître les dégâts. »
"Six questions sur l’incendie de Notre-Dame de Paris", Le Parisien, Jean-Michel Décugis, Vincent Gautronneau et Jérémie Pham-Lê| 15 avril 2019, 23h40
Selon les premiers éléments de l’enquête, l’incendie aurait démarré dans les combles de la cathédrale. Le feu se serait propagé très vite sous l’effet du vent, dévorant l’une des plus anciennes charpentes de Paris constituée de centaines de poutres de chênes. La toiture de plomb de plusieurs centaines de tonnes qui reposait sur cette « forêt » est partie en fumée. La flèche haute de 93m qui surplombait la croisée du transept s’est effondrée en moins d’une heure. Sous la charpente, « les températures ont pu atteindre 2000 voire 2500°C, une température bien supérieure à la celle de la fusion du plomb », explique Guillaume Legros, enseignant-chercheur à l’Institut ∂’Alembert et ancien doctorant de José Torero, spécialiste mondial des grands incendies.
"L’incendie de Notre-Dame de Paris", Sorbonne Université, 19 April 2019.
Most sources seem to quote a temperature of 1000 °C for this incidence, but other sources even go up to 1400 °C:
Fires peak at 1,400°C, explains professor Guillermo Rein, the head of Imperial College London's fire-studying Hazelab.
Nicole Kobie: "The hot, dangerous physics of fighting the Notre Dame fire", Wire, Tuesday 16 April 2019
Interestingly, the place where the cross is located is far away from where most of the flames ravaged, beneath a stone vault that remained largely intact. But nevertheless the wood that did burn on and in the roof was well seasoned, old, and dry.
But even fresh wood that would otherwise make for quite a suboptimal fuel burns often hotter than 600 °C:
Q. At what temperatures do forest fires burn?
An average surface fire on the forest floor might have flames reaching 1 meter in height and can reach temperatures of 800°C (1,472° F) or more. Under extreme conditions a fire can give off 10,000 kilowatts or more per meter of fire front. This would mean flame heights of 50 meters or more and flame temperatures exceeding 1200°C (2,192° F).
(Natural History Museum of Utah: "Wildfires: Interesting Facts and F.A.Q." .PDF)