Not to take anything away from zentaras answer, Pane very likely is a better way to go. However, it seems to me just adding a "raise" to your 'Put text in frame' button command will make it do what you (or I, at least,) would expect.

$top->Button(-text=>'Put text into frame', -command=>sub{
    if(!$frame){return};
    $text->packForget;
    $text->pack(-in=>$frame);
    $text->raise; #<=== Add this line
})->pack;
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