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Doctor Strange seemingly has the ability to open portals wherever he wants. We see at the beginning of Infinity War what happens when he closes a portal and someone isn't fully through it.

Is there any reason why he can't open a portal around the left-elbow of Thanos, and then close it?

I realize that Thanos probably wouldn't allow this to happen normally,

but when he is mostly asleep, and Tony and Spider-Man are pulling on the gauntlet, Doctor Strange has ample opportunity, and plenty of space to just sever Thanos' arm.

Why doesn't he?

Not everything or everyone is equally tough

Fenrir’s teeth can penetrate Hulk’s skin (and that’s cinematically treated as pretty surprising), but when Bruce Banner tries to shoot himself in the mouth with a bullet, it will just bounce off the Hulk’s digestive tract.

BRUCE: I got low. I didn’t see an end, so I
put a bullet in my mouth and the other
guy spit it out!

_{The Avengers}

I think this demonstrates fairly well that “flesh” is not just one thing in the MCU. Some creatures are far, far more durable than others, and Thanos is probably near the top of that list. The flesh of a mutated berserker or an alien god are very different from the body of a regular human being.

Just because some of Thanos’s minions are fragile enough that a portal can fairly easily sever their limbs, doesn’t mean that Thanos himself is.

This is a common fantasy theme, particularly in comic books. On Supergirl, for instance, regular bullets might just bounce off of her, but an energy axe is a different matter.

Are we to assume that a sling-ring portal can sever anything, no matter its durability? Hulk, Thanos, even the Infinity Stones themselves? The implicit assumption in the question is that the sling portals simply “cut through” anything, that there’s no force involved in their closing, but we don’t have anything to suggest that this is the case, and a lot to suggest it’s probably not.

The Power Stone might have increased his durability even further

Recall that in Guardians of the Galaxy, the Power Stone greatly increased Ronan’s durability. To quote the Wiki:

Similarly to Malekith while he was wielding the Aether, Ronan’s
durability was enhanced while bearing his Power Stone-infused
Cosmi-Rod. He withstood a direct, close range shot from the Hadron
Enforcer, with no apparent injury as well as not being at all affected
by the crashing of the Dark Aster as he walked away from the massive
wreckage of his ship as if he had landed without problems.

The Power Stone was the first stone that Thanos acquired. Whatever his ordinary strength and durability, it would have been increased beyond all reason. And, as noted in the description, the other stones may have increased Thanos’s resistance to damage as well.

Strange had tried other things

As noted in the other answer, the Time Stone has allowed Doctor Strange to see all the possible outcomes. All we can say is that something would have gone wrong if they’d tried that. Perhaps, indeed, the thing that would have gone wrong would be that the portal wouldn’t be able to sever Thanos’s skin before he could react, but perhaps any number of other things would have gone wrong.

Because Strange has seen over 14 million combinations and variations of the future and they lose in every one of them but 1. It is the writers way to say: I know what you're thinking, but it has to be like this. Because we, the Fate, say so.

Presumably, he had tried that exact thing in at least one of the variations and it had failed.