The question is, how flat do I need the glued faces to be?
Since you're joining face to face pretty flat is good enough.
If you were joining edge to edge it would be more critical, because boards are stiffer in that direction and it's harder to deflect them when clamping to account for any unevenness. But boards readily bend in the opposite axis, so flattish will usually be fine because only light clamp pressure is needed to close up any minor gaps1.
Something to bear in mind is that many people doing these sorts of laminations with 2x material don't plane the surface of the boards they've bought, they don't even sand them. It's quite common for boards to be glued together after being cut to approximate length and no other prep work. This isn't the way to ensure a good result, but lots of benches have been made this way and it's a testament to how good glues are that many of them have remained solid despite the shortcomings of the construction method.
So essentially what you're doing with your hand plane is cleaning off the surface of the boards (a good thing) and barring any major twists or bows2 you could be good to go after a light skim with the plane.
I'm measuring with the corner of my plane across the boards at various points along the length and looking at the light that makes it through.
You hopefully won't need them here but for future reference you could do with having a long straightedge and a pair of winding sticks for checking for flatness and wind (twist) in boards that you're planing. Even more so for glued-up panels.
You can buy commercials versions of both but they are easily made by hand in the workshop, in fact they have often been given in woodworking guides as good early projects for the learner woodworker. You can make them from plywood for dimensional stability and resistance to warping (use good ply) but both can still be made from solid wood as they used to be. Select good straight-grained stock, with no knots and ideally with rift-sawn or quarter-sawn grain3.
Once you select the wood for the qualities mentioned the species becomes less important, even decent pine could be used. But they'll last longer and remain in better condition if made from a tougher softwood or from hardwood. After completion application of a good finish is advisable to help them remain stable.
Or if you don't care about them being made from wood you could just use lengths of aluminium extrusion. Aluminium is easily cut with any modern saw.
1 But use higher clamp pressure when you're doing the glue-up! You must use high clamp pressure to ensure tight, strong glue joints if using PVA glue of any sort (white or yellow).
2 Which ideally you shouldn't. It best practise to select boards carefully at time of purchase to avoid any with these defects (and others, including too many knots), even if this means having to buy from more than one location or building up stock from repeat visits over a period of time.
3 The wood along one edge of a wider plain-sawn board will often be found to have suitable grain that's close to vertical, saving having to buy rift- or quarter-sawn wood and pay a premium for it.
