He dropped it when Edwin Hubble showed that the galaxies are moving away from us and that the universe is expanding. At that point the notion of a constant value that imposes a static universe became untenable.

Note that it wasn't so much that the constant as such became untenable, but if the purpose of introducing it is to create a static situation and the situation is showed not to be static, then the reason for selecting the given value goes away. If you no longer have a reason to set any given value, you have to consider whether you need it at all.

Although the "history" in physics pedagogy is often made up, IIRC it is just a case of him trying to enforce a static universe.

He added it to enforce a static universe, which was the predominant viewpoint at the time (more rooted on theology than experimental evidence). Then he dropped it and called it "his most embarrassing mistake" when Hubble showed the expansion of the universe.

Finally, our current models put it back in to explain the accelerated expansion of the universe.

Einstein definitely made a mistake but his mistake was not adding the cosmological constant, it was adding it for the wrong reasons. He should have added it as a free parameter to be fitted against empirical data.

It wasn't theology at all. It's just that he was equipped with a less rigorous analogue of the Raychaudhuri equation and an intuition about the focusing theorems one could extract from it. In short: a universe with reasonable distributions of matter in it (galaxies) would collapse to a point eventually. Since the Earth was billions of years old (with improved estimates growing in age), the universe must be older still, yet it had not collapsed. Either there was a lot less matter in the universe than there appeared (and again, better observations kept finding more "nebulae" and then galaxies) or something acted against the tendency to collapse. A simple scalar multiplier on the metric tensor was a straightforward and simple way to proxy for the latter. "Static" in this context simply meant "non-collapsing over billions of years", with the value depending on the averaged matter-density.

Later he and de Sitter discovered that the multiplier could be set to zero (and consequently omitted from the field equations) while allowing an arbitrarily long -- eternal, even -- expansion of a hot big bang universe, even for arbitrarily large average matter-densities.

It wasn't until decades after his death that better observations demanded the return of the small positive cosmological constant. This time it wasn't to prevent total collapse, but rather to account for the gravitational drag of early galaxy clusters on each other not slowing the Einstein-de Sitter expansion (and even later speeding the expansion up).

The cosmological constant term was always a free parameter. It's just that there was no point in writing it down if it was measured as zero.

Think of the accolades if he'd correctly predicted expansion

To this day we would still be talking about him.

The problem is that, given relativity, he could have equally predicted collapse. There was really no reason to think it would go either way. If he'd said "the universe therefore must be expanding" he'd have been unjustified.

Several people -- and the article at the top -- talk about a "static" universe.

What this means is that the universe doesn't collapse into a point, even though it is filled with galaxies. Something had to offset the long-term tendency of masses to gravitate towards one another, and he introduced the most simple mathematical object to represent a plausible "something".

When it became apparent that distant quasars were moving away from our galaxy, he recognized that simple inertia from forces exerted on quasars in the distant past could push an eventual total collapse arbitrarily far into the future. This meant that the cosmological constant could be set to zero without observational conflict, and so he simply removed it as superfluous. He and a collaborator formulated the Einstein-de Sitter universe[1], which is a model of a universe that expands forever with zero cosmological constant. The model remains useful in teaching cosmology, as it is a good representation for the history of the universe back to the formation of the first galaxies to today.

Observations of the Cosmic Microwave Background after his death raised difficulties for the Einstein-de Sitter model, but it wasn't until the accelerating expansion of the universe was detected in the late 1990s that it became necessary to add a non-zero cosmological constant back in. In essence, if an Einstein-de Sitter like universe had galaxy clusters flying apart at some threshold speed they could still fly apart for arbitrary zillions of years, however they would still exert a drag on each other, particularly early on when they are relatively close together. No such drag can be found, so observations do not require explanations more complicated than a small positive Cosmological Constant on an otherwise Einstein-de Sitter universe (and most such more complicated explanations predict things we should see but don't, or mispredict things we do see).

I don't know why he called it a blunder. That seems overly dramatic. He thought evidence compelled an adjustment to his original field equations to avoid the universe collapsing; later observations made him think that adjustment was unnecessary, so he removed it. He died before even later observations made everyone else working in physical cosmology think that first adjustment was necessary after all, to avoid the universe's expansion slowing down at early times more than it did. That happens a lot to all sorts of physicists if they work in a particular area long enough, and Einstein himself proposed, modified, and abandoned numerous other models and theories over the course of his long career.

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[1] encyclopaedic and reasonable, rather than overwhelmingly technical : https://en.wikipedia.org/wiki/Einstein%E2%80%93de_Sitter_uni...