Being grappled means the only thing the targeted character can attempt is to break the grapple.
One cannot grapple xenomorphs, but I'd say someone else should be able to attempt the break grapple... We see attempts (that fail) in the films.
I think it's safe to assume a PC can grapple (or resist a grapple) with a face hugger. I always assumed the rules saying one cannot grapple a Xenos refers to the full grown version. A face hugger and a chestburster are small enough that they can be grappled. We actually see this in Aliens as someone else pointed out when Ripley holds the Facebugger back from her face. I'd play that scene as an opposed grapple test. Then when Hicks and Vasquez pull it off that's some kind of assist on the grapple check.
So in the situation the OP mentions, the PC has until the Facehuggers next action to beat it in a grapple check. Should they fail, that's pretty much the end for that character.
I think you're failing to grasp the mechanical meaning of grappled.
The common non-game meaning is that you can grab them. That's all well and good, but that's also irrelevant to what the mechanics mean.
The mechanical meaning of grappled is "you can only resist grapple." Which also means you cannot attack while grappled. You cannot flee while grappled. You cannot dodge while grappled
The facehugger cannot be prevented from attacking by grabbing it. It's got too many ways to make with the hurt.
The facehugger is able to wriggle away pretty well; you're not going to inconvenience it on dodging unless you have a very specific grip on it; that would seem to be using an unarmed attack to aid one's own or someone else's attack.
The facehugger is also likely able to escape no matter how you grabbed it.
Therefore, grabbing it is not mechanically represented by the Grapple action and inflicting the grappled condition. It's assisting, or unarmed attacking with a second success to provide a benefit against it, or giving it a shove.