Briefly, the 1e system had the following problems:
1) It was not always clear, and very easy to become confused, regarding which saving throw was applicable in which situation. If a spell could kill you, did you save versus spells or death magic or was the death magic category only for non-spell death effects like a Bodak's gaze? If a spell could turn you to stone or change your shape, did you save versus spells or petrification/polymorph? What about a wand of polymorph? Some have argued that if more than one category was applicable, you always took the leftmost column. If that was the intention, it was not exactly made clear and isn't at all intuitive.
2) It was not clear what saving throw was applicable in a novel situation or versus an unclassified effect. For example, although it can be argued that the logic of 'Rod/Staff/Wand' saves was based on the idea of which class was best able to dodge a ray, it was not immediately obvious that the logic of the save versus 'Rod/Staff/Wand' was based off the idea of dodging a ray aimed at you. Therefore, many DM's might not pick up on the fact that 'Rod/Staff/Wand' was the ideal save for a similar idea of dodging away from something. This can be seen in 1e modules, where frequently when saves were called for in similar situations, different designers even of published official TSR material would call for very different sorts of saves: save vs 'rod/staff/wand', roll under dexterity (semi-formalized as an 'ability check'), straight percentage chance of avoidance, etc.
3) While using the saving throw tables is fairly easy in the best case, analyzing them is actually highly difficult. This is because classes did not gain levels at an equal rate in 1e, and so an actual analysis has to be made on the basis of how much XP earned grants what sort of saving throw bonus in which categories. This is a 3 dimensional array of not so easy to recognize divisions and once you do actually do this analysis it becomes clear that the saving throws really aren't that well thought out if the idea is to produce some sort of balanced intuitive system that grants predictable advantages depending on the archetype of your character. In particular, there is the usual masochism if you want to play a thief of being significantly worse than every other class at just about everything that seems to have been largely ignored until 3e.
4) 'Spell' was a very catch all category, but it isn't at all clear that if the logic of the system is based on the sort of a effect you are saving against that it should be. If 'petrification' is better thought of as being its own category, or 'death magic' is better thought of as being its own category, why shouldn't 'fireball' or 'cone of cold' be more like saving versus a 'breath weapon'? I mean why save versus Rod/Staff/Wand when the wand shot a fireball at you, and against spell when a spellcaster did? What's the logic of that? Why save versus 'breath weapon' when a white dragon breathed a cone of cold at you, and 'spell' when a caster cast a 'cone of cold' at you? Aren't you in game experiencing more or less the exact same effect?