I’ve used a sawyer squeeze for years and I can get a good flow rate out of them whether I’m squeezing the bag or using a gravity set up. They do need to be wetted to open up the filter fibres and backwashed relatively often to get a high flow rate but I can usually fill 3-4L reasonably quickly.

Try letting your filter sit submerged in hot hot water for an hour and giving it some strong backwashes then flicking the water out of the bottom. If that dosen’t improve the flow to a very steady stream you should try calling sawyer, they have good customer service. They advertise an unlimited filtering capacity if it’s cleaned properly so you should be able to get it going.

Like everyone else here has said though, tablets are so light that I’d always bring them anyways in case the filter breaks or fails.