lemmy seems to have lost my response to this, so I'll type it again and hope that it doesn't show up twice

There's three separate issues here:

1. The ability to express multi-sorted algebras

2. The ability to extend algebras with new sorts

3. The ability to extend a single sort of an algebra with new variants

For point 1, object algebras do support this even though I didn't show it in the post:

interface ExprAlg<Num, Bool> {
    lit: (value: number) => Num;
    add: (left: Num, right: Num) => Num;
    eq: (left: Num, right: Num) => Bool;
    iff: (interrogee: Bool, then: Num, els: Num) => Num;
}

const evaluate: ExprAlg<number, boolean> = {
    lit: (value) => value,
    add: (left, right) => left + right,
    eq: (left, right) => left === right,
    iff: (interrogee, then, els) => interrogee ? then : els
}

function makeExample<Num, Bool>(alg: ExprAlg<Num, Bool>): Num {
    return alg.iff(
        alg.eq(
            alg.add(alg.lit(2), alg.lit(2)),
            alg.lit(4)),
        alg.lit(1),
        alg.lit(2))
}

console.log(makeExample(evaluate)); // prints 1

For point 2, you are correct that the original Java formulation of object algebras does not support data sort extensibility. I haven't tried to see if TS's more powerful generics change this or not.

For point 3, consider what happens if you want to add a new variant to the data type. In this case, add a Mult variant for multiplication. With GADTs this is not possible without modifying or duplicating the existing evaluation code, but I can do it with object algebras:

type ExtendedExprAlg<Num, Bool> = ExprAlg<Num, Bool> & {
    mult: (left: Num, right: Num) => Num;
}

const extendedEvaluate: ExtendedExprAlg<number, boolean> = Object.assign({}, evaluate, {
    mult: (left: number, right: number) => left * right
})

function makeExtendedExample<Num, Bool>( alg: ExtendedExprAlg<Num, Bool>): Num {
    const one = alg.mult(alg.lit(1), alg.lit(1));
    return alg.iff(
        alg.eq(one, makeExample(alg)),
        alg.lit(3),
        alg.mult(alg.lit(2), alg.lit(2))
    )
}

console.log(makeExtendedExample(extendedEvaluate)); // prints 3

This is the point of object algebras. ADTs and GADTs can't do this out of the box; this is the essence of the expression problem and is why more advanced techniques like final tagless encodings and datatypes a la carte exist.