The globe always explodes within 1 minute
The core of your player's argument is that the last sentence, about the globe exploding after 1 minute, is intended to be contingent upon the previous sentence describing "setting down" the globe. I don't think this is a reasonable reading of the spell. The sentence about setting down the globe is itself a clarification of the previous sentence about throwing or slinging the globe. The purpose is to clarify that it is the force of impact from being launched at high speed that shatters the globe, not simply any contact with the ground or an obstacle. Basically, as long as the globe is treated like an egg, it won't shatter prematurely. None of this has anything to do with the 1-minute time limit.
So, by my reading, the final sentence of the paragraph stands alone. Regardless of what you do with the globe, it always explodes after 1 minute unless you detonate it early. (If I were rewriting this spell, I would put this sentence in its own paragraph to make this extra clear.)
Balance dictates that stockpiling numerous freezing spheres is unintended
D&D is balanced around the idea that the party has limited resources, many of which (including 6th level spell slots) are only recovered on a long rest. Allowing players to stockpile a large number of 6th level spells to be used all at once badly breaks that assumption, and will throw off the balance of the game as a result, allowing players to deal 10d6 cold damage every round for many rounds in a row, as long as they have had sufficient time to prepare, or even allowing a party of 4 to deal 40d6 cold damage in one round by all slinging spheres at the target.
Also notable is that most (all?) spells that do allow indefinite "stockpiling" come with significant limiting factors to prevent exactly this sort of thing. The two that come to mind are Glyph of Warding and Symbol, which are used to set triggered traps. Both spells consume a costly material component with each casting, and the resulting trap cannot be moved from where it was set. The fact that Freezing Sphere has none of these limiting factors is further indirect evidence that "stockpiling" is not an intended mechanic for this spell.
(Also, even if you did allow the 1-minute detonation timer to only start once the sphere is "set down", your wizard is going to have a tough time taking long rests without dropping their bag of live ice grenades.)
You can stockpile freezing spheres... for up to 9 rounds
However, assuming you have the spell slots to do so, you can cast multiple Freezing Spheres, hand them out to your party, and toss them all in one round at a target you really hate to deal massive damage to it, all within the span of one minute. This could be a viable tactic to open an ambush against a very tough opponent. (Make sure the entire party has slings, or make sure you can throw the spheres from behind cover, or you'll get caught in the blast.)