Consensus in the autocross world is the newer Azenis 615 side wall is not as stiff as the older 215 they replace, hence the lack of crisp turn-in. Many swear by the Azenis, I have yet to personally experience them but i really am enjoying the set of PS2's I recently put on. But mine are on 17" stockers. My observation has been that tread depth appeared to be similar on each when new. The "squishy" sensation you describe is what we call tread squirm. You may want to experiment with various tire pressures to maximize turn-in feel and minimize the squirm. Now if you have an unlimited tire budget...
you could always have the tires shaved, that would also reduce the squirm...
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