It's weird that folks are making a fuss over things that happen when PCs are "off screen" but are fine with the presentation of things as they are upon the PCs' initial arrival.All the influence is very much overstating it. The only influence the players have is chosing not to engage with the guild. What happens instead is completely DM-driven and is largely independent of the players' decisions. Does the guild thrive in the absence of the PCs? Does it struggle due to in-fightings and resistance from the community? Does another band of adventurers deal with it instead? Does the local government institute draconian anti-theft measures that unjustly target innocents and criminals alike? Does the guild raid a long-forgotten vault beneath the city and unleash horrific hauntings?
When and if the PCs return to the city, the plot-wagon selected by the DM will be waiting for them.
I don't have any problem with this process either as a DM or as a player, but I think it's important to acknowledge that player's influence on game world events is largely inconsequential.
I don't think that the world should "freeze" when the players look away, but I also want to acknowledge that the off-screen events do not flow as some natural, living world that runs on it's own. In most D&D games, off-screen events are DM-driven without player influence.
Imagine: the PCs come into town. They spend some time searching for rumors and they discover a couple plot hooks. There is a thieves guild on the cusp of making a major move. There's also a dungeon a few weeks away that promises great riches. After examining their options and investigating both, the PCs decide to head off to the dungeon with full knowledge that by the time they get back, the thieves guild will likely have made its move. They have made no move of their own to interrupt that plan or even involve themselves in it.
That is a choice. That is agency. While they can't know for sure what the consequences will be, they know there will be consequences. It is no less a choice than getting involved.