Sounds like you found a solution!
I mixed up the support request with another regarding the “hidden results’ field” being tab-able – in that case, tabindex -1 did fix that, as far as I recall.
Regarding accessibility and tab order. Have you considered, changing the magnifier icon from a DIV to a BUTTON instead? That would increase the inherit semantic meaning, as well as make it totally tab-able as shown in quick screenshot:
https://paste.pics/48cbead800b4c6d62d08b87e0af793de
I know that in its current implementation, the input field is plaster AFTER the magnifier button, making the tab-order a tiny bit weird (8 and 9 in the screenshot), buuut, that could probably be addressed by outputting the elements in their logical and expected order in the DOM, and if needed, placing the magnifier before or after the input field visually using flex or grid order.
An accessible (as can be) solution is important to me and my visitors as well. I’d be happy to help testing out proposed solutions.
An @wpdreams – accessibility is about progression, not perfection – I appreciate your effort in this area.