RAW it will fall, but it depends on how you want to play
We have been playing it as described in 1., stuck in the webs, which has not caused any issues. The whole point of real webs is to catch flying creatures that then are stuck in place. We treat the magically created webs the same way, and enjoy the verisimilitude.
The strict RAW outcome limited to the flying and web rules would be 2. Spells only do what they say they do. The spell does not say the creature is fixed in place, and neither does the restrained condition, which merely states your speed is 0 (PHB, Appendix A). A speed 0 flying creature without hover or special magic will fall (PHB, p. 101, flying movement). So the strict RAW interpretation is that the creature would fall. By the core rules, you fall straight to the ground, as if there were no webs in the way.
Xanathar has alternate optional rules by which you fall just 500 feet, and with a reaction from a readied action you might be able to resume flying when you use those rules as you leave the web. These questions about falling have already been answered separately in depth, so we'll not rehash all of that here.
The webs count as difficult terrain. Difficult terrain’s rule effect is merely that you move at half speed (PHB, p. 190). By strict rules, falling does not count as movement, so you would fall at normal speed. You could interpret this as the webs being too thin to slow the rate of falling. RAW, they will take damage from the full height they fall.
I would likely ignore that unintuitive outcome and say that the creature is dragged down through the web at reduced speed, and limit potential falling damage to the height of the fall after they left the webs. But we don't have this issue as we use interpretation 1.
The bigger picture
You ask which answer is the "correct" answer. This depends on your goal for the game. There are many different ways to play D&D.
Many of your questions, like this one here, this one about Arcane Hand, this one about line of sight, or this one about gaze attacks share that they are looking for an exact clarification of specific rules mechanic interactions, where these mechanics are unintuitive and lead to a different outcome than what one would expect. It may be that your route to enjoyment is to have such a deep, technical reading of the game rules specific to the matter at hand, and that is entirely valid. If so, the "correct" answer for you is your option 2.
Other variants of this game like D&D 3.5, 4e or Pathfinder might be worth a look for you. They are considered to aim more at providing rigid mechanics than D&D 5e.
The designers of 5e consciously avoided to specify rules for everything and even to define all game terms. The DM is much more encouraged to make rulings that ignore rules. There are multiple explicit statements about this in the rules, which are as much part of the rules as any specific mechanic:
From the DMG, p.4:
The D&D rules help you and the other players have a good time, but the rules aren't in charge. You're the DM, and you are in charge
Or from the Sage Advise rules compendium intro:
The DM is key. Many unexpected things can happen in a D&D campaign, and no set of rules could reasonably account for every contingency. If the rules tried to do so, the game would become unplayable.
Or from Tasha's Cauldron of Everything, p.5.:
You don't need to know every rule to enjoy D&D, and each group has its own style -- different ways it likes to tell stories and to use the rules. Embrace what your group enjoys most. In short, follow your bliss!
This is not to say that one cannot figure out some of these edge cases for specific rules interactions by spending time poring over the rule books (and likely, this is what many of us here enjoy, and it can lead to great hedge philosophical discussions). What the game tells you is that it is not that important to then adhere to what you find, if you do not like it.
For a group that enjoys imagination and picturing the situation more than rules mechanics, it is entirely valid to ignore rules that would contradict their expectations. This is as "correct" a reading of the overall body of rules as the more narrowly focused, technical answer about the specific mechanics involved.
If you want your magic webs to stick flying creatures to where they are, the rules support that by encouraging you to ignore or override them.
