Allright, so I was thinking about this, so here I go...
Actually, I normally have the "Remember" option off personnaly. Yet if I close XY with many Finding tabs, they'll be there when i start it over and each of them will remember its own search settings, as long as the active tab was NOT a Finding tab, but a Browsing one.
The only case in matter here is when the ACTIVE tab was a Finding tab. Especially because then, XY do the search on its startup, aka when still hidden, so one might thing he's not starting up or something as we have no indication to tell it actually is.
So maybe, and this looks like what j_c_hallgren suggested, make it that XY does his startup as now but, special case where the active tab is a Finding one: then show XY first, and
then start the search. Means no more need for a "whats a good name for this one" option

since one can see XY is doing a search, and also can press ESC to stop it anytime!!
What you think of this?
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Little Offtopic: I think you got a bug somewhere with this. I was doing some tests, and I have now both "Remember" & "Save settings" enabled. I have a few Browsing tabs, and 3 Finding one. Each of thoose has different search results.
When I start XY, after closing it with a Finding tab as active tab, it will redo the search but NOT in the tab where it should be, instead it will use the Option setting (current, new tab or SR tab), which can lead to loosing a search (setting to SR tab makes it loose the SR tab results, overwritten by thoose of the "remembered" tab); or opening a new tab on startup with the results, leaving the tab they should have been in shown as Finding tab (icon), but activating it turns it into a Browsing one.