Plaguescarred
Hero
While a DM can always rule otherwise, the surprise rules as written are not automatic, instead offering a chance to not be surprised by noticing enemies using Stealth vs Perception contest. Which what you say doesn't offer any chance.But the spiders don't have enter the material plane until after initiative has been rolled, so the PCs see no threaths at the start of combat and are thus surprised.
Scenario 1: If initiative is rolled before the phase spider emerge, the PC may loose or win it but not act before the spiders anyway, not knowing about their presence. When the spiders emerge on their turn the PCs would notice them not being hidden and be able to take reactions if they lost initiative where they wouldn't if they were surprised.
Scenario 2: If initiative is rolled after the phase spider emerge, the PC may loose or win it and act before the spiders, knowing about their presence. When the spiders emerge at the start of combat the PCs would notice them not being hidden and be able to take reactions if they lost initiative where they wouldn't if they were surprised. In this scenario the spiders were allowed to take an action to emerge before initiative which is not RAI as per Sage Advice http://dnd.wizards.com/articles/features/rules-answers-november-2015
"If anyone is surprised, no actions are taken yet. First, initiative is rolled as normal. Then, the first round of combat starts, and the unsurprised combatants act in initiative order."
So bottom line is PC might still possibly be left unable to act before the spider emerge, but in either case it would leave them with the possibility to react.