You are correct. I think OP was asking why trade throughs happened, so I was thinking in terms of liquidity takers. The hidden orders which improve on the NBBO are implicitly protected because there's no reason for exchange to route the crossing order out when they can match it internally. So for passive orders you don't need to be concerned ... You will not be traded through. For aggressive orders, you need to always be concerned as you maybe chasing stale NBBO.
Roughly speaking, a marketable order is one which can be filled immediately, because it is priced at the current market price or better. (Better here means higher than ask for a buy or lower than bid for a sell). Market orders are always marketable unless there is no quote on the exchange.
That's not what protected means. Hidden orders can be traded through on other exchanges than the one where they are resting. Visible orders can't, except by special exempt order types (e.g. intermarket sweeps)
I think we are just looking at it from different perspectives. So from a trader's perspective, if I place a resting displayed/lit order on an exchange, am I guaranteed not to be traded through? Yes, it's protected. If I place a hidden order in between the inside market, is it guaranteed not to be traded through? Yes, it's guaranteed to be filled prior to lit bid/offer is taken out. Agreed? The hidden orders placed (or ended up being) outside of NBBO is not what I was referring to.
No. Hidden orders are not guaranteed to be not traded through as they are not protected. What you said is only true within the context of a single exchange. For example if I have a hidden buy order on exchange A, exchange B will happily fill orders below my limit price because B isn't aware of hidden orders on other exchanges. If the order was visible, B would be forced to route out to avoid a trade through.
You are correct that hidden orders are not protected and your example is valid. My point was that hidden orders inside the NBBO are very likely to fill prior to your lit orders, so even though they are not explicitly protected, they benefit by the protection of the lit orders ... That's the whole point why pegged and hidden orders are popular. You are using a more aggressive order to jump in front of lit queue, receive some rebate, and will rarely get traded through the NBBO(most major exchanges are part of the NBBO). So that's what I meant that for liquidity providing orders, you don't need to worry about trade-throughs. Does it make sense?
Is it really true? Why would the hidden orders inside NBBO be very likely to fill prior to lit orders and how is it related to lit orders being protected? A simple smart order router that routes to the best price exchange will only have a 1/N chance of hitting you assuming all the exchanges are at the same price. I think the other party would have to be actively sweeping for hidden liquidity (e.g. send fill or kill limits to all exchanges) to give you a good chance of getting filled before lit orders. HFTs might be doing this all the time, to some extent, but if your limit price is aggressive enough to be arb'd like that, then it might not be a great price and it would be more likely to happen after you were traded through.
Hidden buy order above Best Bid is jumping the lit queue, right? So before Best Bid is taken out, your order will fill (let's assume all major exchanges are at Best Bid). Obviously, if best bid moves, you need to adjust. Basically, the lit quote is something you can use to make decisions knowing it is protected and will be routed to. Exactly ... part of "smart" routing. I think you are thinking of HFTs as some parasitic arb seeking conglomerate, but SOR is part of HFT and may be used to provide client with best execution depending on what they need. There are no "best" way to find liquidity, it depends what it is you are trying to accomplish. Keep in mind that every instrument is different, so what works for liquid stocks may not work for mid-cap or low float. There are no one size fits all solutions.