This is one of several flaws with the execution of the Gate Bridge plot.
One is that the bridge, despite clearly being a forwarding mechanism initially†, turned into a one-shot pass-through later on in the series: ref. "Adrift" in which Apollo dials Midway from within Pegasus which, if it were feasible, would render the entire purpose of the gate bridge more or less moot;‡ the show began making a ton of seeming inconsistencies from this episode forward, as part of its "new direction".
Having the two gates served a vital dramatic purpose in emphasising the "midpoint" nature of the station, but ultimately it makes no sense whatsoever (Pegasus gates and Milky Way gates can dial each other without any problem at all) and required McKay et al to actually write explicit work-arounds for the problem it caused: which gate do you end up at, when both work perfectly well? In fact, even to "come up with a workaround" stretches credibility more than a little given the nature of wormholes snapping to the first charged gate they see.
- Out-of-universe answer? The writers playing fast and loose with logic;
- In-universe answer? The SGC's best and brightest playing fast and loose with logic.
† Hence "no need for an iris or shield" — it would in fact be dangerous to add one, since you'd never be able to get synchronous confirmation that your IDC had been accepted before stepping through that first gate. Presumably this is why, once the security weaknesses of the forwarding macro were discovered, Midway was never rebuilt and the bridge gates were cannibalised for other use (such as Atlantis's new gate in "The Lost Tribe"). A massive waste of time and expense overall, if you ask me, though it felt like a very cool story.
‡ If you could directly dial Midway then, at worst, you'd have to send your travellers from Atlantis to a planet at the edge of Pegasus so that they could "reach" Midway; this still seems a heck of a lot more convenient than placing umpteen Stargates across a galactic void, and then you could have an iris/shield and prevent all that nasty invasion business. The only alternative is that "dialling the bridge" automatically opened up all the gates en route so you could radio Midway, but this is total nonsense unless you have two gates at every single point along the route. The gates receive you, store you, dial the next gate then send you on. No radio transmission to/from Midway, at least not a real-time conversation, can possibly work in this scenario.