This recurring question-affirmation does not have a solid foundation, it is based on an in-depth historical knowledge and is "urban legend" or pure slander.
It has already been said that the Hanbury’s also closed the gardens in the summer because they were not very pleasant. At the time of Thomas Hanbury certainly the large trees that can be admired today did not exist; at the time, most of them were at most 40 years old and there were no such over 100 years old individuals. It cannot be forgotten that the intentions of the founder were above all to acclimatize and experiment with the introduction of a very large number of species, rather than to produce valuable landscape paintings; between the late nineteenth and early twentieth century therefore the GBH were above all an efficient company complex of nurseries and crops, rich in species and including a stable, chicken coop and mills. Subsequently, between the two world wars, the garden underwent profound transformations so that for several periods some sectors had the appearance of construction sites full of activity. Only at the end of these works new avenues and sections that modified the original systems could be observed. During the Second World War the Italian and German military occupation, the minings and bombings had almost completely destroyed the heritage of the gardens. In the 1950s, the Touring guide reports that only with difficulty a daredevil visitor could venture from the entrance to the Roman road. In the 1960s, after the purchase by the Italian state, it took three years to reach (simply reach) the beach from the main access. 
Yet, in the 1970s, gardeners had to pay attention, when hoeing, to possible explosive devices that still lay in the ground.
In addition to all this, we must consider the different financial and personnel resources that Thomas Hanbury enjoyed. In GBH, 40 to 60 gardeners worked from sunrise to sunset; when there was a need, a hundred men were called workforce from neighboring countries, to complete large works in a short time, with daily wages for just the days needed. Even today some parts of the GBH (the lower part to the west) are subject to interventions that tend to restore spaces that can be visited according to the original project, but the current number of gardeners is only just over 10 units.
Ultimately the answer lies in another answer. Are you sure it was so? The memories of writers (for example Nico Orengo) who visited the GBH before and after the state acquisition did not reveal any substantial deterioration and, in general, appreciated the most recent conditions of the GBH themselves.