Does the Clone spell make someone effectively immortal?

While reading through PHB 5e spell list, I found an interesting spell: Clone.

This spell grows an inert duplicate of a living creature as a safeguard against death. This clone forms inside a sealed vessel and grows to full size and maturity after 120 days; you can also choose to have the clone be a younger version of the same creature. It remains inert and endures indefinitely, as long as its vessel remains undisturbed.

At any time after the clone matures, if the original creature dies, its soul transfers to the clone, provided that the soul is free and willing to return. The clone is physically identical to the original and has the same personality, memories, and abilities, but none of the original’s equipment.

One might assume that dying of old age triggers the spell, and that the character is restored to life in a new (younger, of course) clone. That would make anyone with access to that spell (either by being a 15th-level wizard or by having such a wizard nearby) and a little bit of gold (3k, really? And if the wizard is 17th level, it isn’t even a problem) effectively immortal, right? And, if we were talking about the real world (or at least some fictional world where such thing is explicitly available by design, such as EVE Online’s world), than the answer would probably be YES.

But, we live in the cruel world of D&D, where characters can have this ability:

At 15th level, your ki sustains you so that you suffer none of the frailty of old age, and you can’t be aged magically. You can still die of old age, however. In addition, you no longer need food or water.

…and can, as being said, still die of old age without ageing. So the question is not really as simple. There is a point suggesting that such a scheme of constantly recloning yourself each time back to your younger self might work by RAW:

  • All spells, that somehow restore the dead back to the living (like Resurrection) have an explicit exception defined, that this spell cannot restore life back to someone who died from old age. The Clone spell does not have that kind of exception, however. So, by the basic D&D rule of Specific beats General, it seems that any spell that would restore life back to dead without such an exception would be able to do so even if they would die from being too old.

So, to summarize my reasoning, the question bothering me is this:
Does the Clone spell allow someone with access to it to be able to live forever?

P.S. If I self-answered the question, I apologize. It just seems to me kind of confusing and I would like to hear some other opinions to understand the designer intentions behind this spell.