How would changing the timing of the "starter" affect strategy/tactics?
There are edge cases where it matters. If 


you threw a card in your opponent's crib
that card came up
your opponent is close to winning
you are also close to winning


You would consider making more aggressive plays when scoring to try to out-peg your opponent. You may open yourself up to your opponent scoring more off your aggressive plays, but it's a risk you have to take because if you don't then you expect with high probability your opponent will win.

If you concealed the card, you wouldn't have that information about your opponent's crib during the pegging hand.



Also, if a Jack comes up, that player gets points and can win on the spot if they're close enough. (I forget if it's one or two pegs... but assume they're on the last one to win.) If you are close to pegging out too, you might win on pegging when your opponent would have won if the Jack would have given them the win.