If a hashing algorithm is insecure, doesn't that make it a great compression algorithm instead?

No but it might be a good compassion algo.

Technically a hash that is reversible on 20% of inputs is a lossy compression algo; you can recover 20% of the input in a statistical sense.

Hash functions are rarely fully reversible, usually a weak hash will either reveal partial information about the input or reveal only some inputs. Technically, if you have a massive lookup table, a perfect hash is a perfect compression algo.

Hash functions are not reversible by definition.

Flaws in desired properties such as uniformity or collision probability can be found, all right.

But that still doesn't allow you to retrieve the full original message.

It can break some security schemes, but that's because of how hashes are used.

Lossy compression and hashing kinda overlap, but are not the same.

A good hashing function is greatly sensitive to very small changes in input (non locality property), whereas lossy compression shouldn't be. Small changes in input should result in perceptually similar output.

Are all hash functions lossy compression? Mathematically, yes. (You could have higher cardinality on the image space, sure, but then I guess it's not really a hash function).

But neither every hash function makes for suitable lossy compression, nor every lossy compression algorithm makes for a secure hash function.

In fact, they are polar opposites.

The more "continuous" (in the mathematical sense) a hash function is, the less suitable it is as secure hashing, but the better it is at lossy compression.

And given algorithms are researched to improve on their expected usage, yeah, no, there's no overlap, save for certain mathematical technicalities.

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Yeah, hash is about as reversible as recovering X from

X % 3 = 2
recover X....

You can use attacks on some weaknesses and dictionary comparisons if you know the data is at least actual language (or some other known final form) but that's about it

@Hazarth ask one of those recovery wizard hackers 😂