##100% RAW##

I would agree that

The target can move only by pushing or pulling against a fixed surface, which allows it to move as if it were climbing.

means that the target can no longer fly and can only move using fixed surfaces.

##My interpretation##

I would personally rule that the target can still fly because they are using some force to move, either pushing the air or with magic, just as they would do if they pushed off a solid surface.

100% RAW

I would agree that

The target can move only by pushing or pulling against a fixed surface, which allows it to move as if it were climbing.

means that the target can no longer fly and can only move using fixed surfaces.

My interpretation

I would personally rule that the target can still fly because they are using some force to move, either pushing the air or with magic, just as they would do if they pushed off a solid surface.

##100% RAW##

I would agree that

The target can move only by pushing or pulling against a fixed surface, which allows it to move as if it were climbing.

means that the target can no longer fly and can only move using fixed surfaces.

##RAI##

I would personally rule that the target can still fly because they are using some force to move, either pushing the air or with magic, just as they would do if they pushed off a solid surface.

##100% RAW##

I would agree that

The target can move only by pushing or pulling against a fixed surface, which allows it to move as if it were climbing.

means that the target can no longer fly and can only move using fixed surfaces.

##My interpretation##

I would personally rule that the target can still fly because they are using some force to move, either pushing the air or with magic, just as they would do if they pushed off a solid surface.