When you buy an option on SPY <snip> incidentally can be exercised any time.
I don't think "exercised at any time" is an incidental point -- it's the whole point.

Because you can exercise an American style SPY option at any time, then a SPY ITM put's time value (aka extrinsic value) must be non-negative. Otherwise you could buy 100 shares, buy an ITM put contract, and immediately exercise the ITM put to sell the shares at strike (which is higher than the share price for ITM puts), and make "free money". The 'ask' will quickly rise to stop people making 'free money'.

But a European option has no early exercise, it's cash settled on the expiration date, hence the above process doesn't apply.

I haven't worked out an arbitrage argument using futures to show this, but I'm betting that the resolution to the examples I showed where ITM SPX puts have "negative time value" is:

What is relevant at any time before expiration of a European option is not the price at that time relative to the strike, but rather the price at that time relative to the discounted value of the strike.
By 'discounted value' I mean the value you'd have to put into a risk free asset today to have the value of the strike at expiration. The broad point (that admittedly I haven't fleshed out) is that the only thing European options care about is expiration. So, if you want to talk about "extrinsic value" or whatever before expiration, then you need to discount stuff back from expiration.

I tried the 'discounted strike' idea on the examples that had negative extrinsic value (time value), and it fixed them.