Yes, cloning, repeatedly, maintained, could make you immortal.
There's some logistical concerns that make this trickier than the spell itself:
Vessel must be undisturbed
So, ideally, you set up a nice young version of yourself, hide it away for the time something goes wrong and go about your life, right? Well, the longer it's around, the more likely, over time, something COULD happen to it. Especially in a world where you've got things like purple worms, umber hulks and bulettes and other critters that dig through granite like butter.
Well, then it makes sense to set up some defensive measures, right? Traps, spells, etc. But in the world of D&D, the more defensive measures you put up, the more people assume it's got something valuable to steal...
Now, as a GM I wouldn't just automatically assume something is going to happen, but if the clone is sitting around for decades, or the wizard in question has enemies seeking them out, then we'd start having to think about problems.
A giant diamond
So, the diamond is worth 1,000 gp. This doesn't mean you can simply pull out 1,000 gp and find these diamonds anywhere, everywhere. There's got to be a limited number of them. You're probably not the only caster who is looking into this spell.
So, a bunch of wizards want immortality from a limited resource and are all looking for it.
I'm sure that's not going to lead to problems.
Welcome to the Immortal Club
So, if you manage to live far beyond even what most D&D world folks know people to be capable of, and you're known to be an awesome wizard... how many other people are going to be trying to get your secret of immortality from you?
How do other things which are immortal feel about this? Do they find a way to manipulate/play you because you're new to this game? Do they already have a control on the 1,000 gp diamonds and dole them out to the few wizards who have Clone just to keep them under their leash?
Is there an alliance of lichs who are jealous you've found a way to live, but actually live, not undead live, and they'd like to simply stomp you down for being audacious?
Are there mind flayers looking to eat the juicy mind of a super-intelligent wizard with 800 years of tasty-tasty knowledge?
Are there divine guardians of life and death who did their accounting and finding there's a soul short that needs to move on?
Gameplay
Unless you're playing a very unusual game of D&D, these issues aren't likely to come up too much simply because the timescale is too short. But it makes excellent source of adventures based on NPCs - just imagine what happens when you do have a wizard who has been doing this and dealing with all of these problems and what that means for the PCs when they get involved in it.
Alive or dead, it stays enlarged for the duration of the spell.
Unless otherwise stated, a spell lasts for its duration.
The only rule that covers general spell duration is a small section on page 203 of the PHB:
A spell's duration is the length of time the spell persists. A duration can be expressed in rounds, minutes, hours, or even years. Some spells specify that their effects last until the spells are dispelled or destroyed.
And the rules that cover monster death aren't very extensive either. Page 7 of the Monster Manual says, under "Hit Points":
A monster usually dies or is destroyed when it drops to 0 hit points. For more on hit points, see the Player's Handbook.
And PHB page 198 says, under "Monsters and Death":
Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws. Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.
Other than that, there's no specific rule that says spell effects end on a creature when it dies, unless the creature is the one casting and the spell requires concentration (PHB, p. 203), so they should carry on (exceptions might be made at the DM's discretion).
Thus, the creature would stay enlarged for the full 1-minute duration of the spell, whether dead or alive, unless your concentration is broken.
Best Answer
There's no RAW or RAI to limit which clone your soul goes to.
Clone provides backup bodies for your soul. You cannot split your soul across multiple clones, but there are no rules or lore about issues with your soul traveling across distances. There's no instance of a soul travelling at a set speed, so there's no benefit, detriment, or limit to picking a clone based on location. The clone spell doesn't even limit you to using clones on the plane on which you died, suggesting the soul could travel through the planes to backup bodies.
In this situation, where there are no limiting rules on which clone can be chosen, there is nothing to prevent a player from picking from any clone that has matured. Souls have been shown to be intelligent enough to make decisions, so the player would retain control over their soul, and it wouldn't be a DM decision (unless you leave it up to them for story purposes).