I have gutted my bathroom down to the studs, and I want to put a shower mixing valve directly where the red X is. The problem is that board outlined in red is load bearing for the staircase landing behind that wall.
I assume I can't cut out a section of that board to fit a valve and reframe the wall without causing structural issues. Is it possible I am overlooking or overthinking this solution?
There are a couple of options that don't involve a structural engineer and major work.
If this is exactly where you want to put the shower, the easiest path forward is to frame out the wall with another layer of studs. You will avoid the huge structural headache that it would cause to try to place a shower valve where that structure is. The downside, of course, is that you will lose 3.5" of the room to your extra layer of studs.
The alternate option is to rearrange your bathroom layout to avoid this structural challenge.
That's effectively a rim joist and you can cut it out such that what remains is equivalent to the studs above and below it. It was only installed as a matter of convenience for the carpenter and isn't required in any way to remain solid.
You want to retain these things as you cut your opening:
Pass-through bearing between the upper and lower studs. You could offset the block for the stud near the X a few inches without concern.
Support for the subfloor on the stair landing. You don't want a completely unsupported stretch of more than say 12".
Support for the joists that apparently intersect this rim, as indicated by the stacks of nail heads. 2" of bearing on the lower wall plate is adequate. It does not need the rim joist for support.
In this case I would cut out a 12" section of just the 2x10 rim, centered on your valve location. Leave the common joist behind intact. From there, assess interference from what's behind and to the sides and ask a new question if necessary.
If it's important for you to put the shower right there, consider a surface mounted shower panel instead of an in-wall system. They are insanely easy to install, you get way more bang for the buck and they are easy to repair and replace without breaking tiles. Here is an example of a cheap one on Amazon so you know what I'm talking about but you can find better ones from plumbing suppliers. It's an especially good fit for your situation.
To cut a huge hole in that board you would have to know exactly how it's being used on the other side. If it's holding up a couple of stair risers or joists supporting the steps, as long as those stair supports transfer into what remains of that board and from there onto the studs beneath, you might be okay. That board is not like a joist, it is supported everywhere. But you can't know that without knowing what's on the other side. [Edit: If you want to go this route, see @isherwood's more thorough answer]
Another possible solution is to use an external shower valve. These are widely used in Europe and are not unattractive. The hot and cold pipes emerge from the wall at a prescribed spacing of 15 cm (6 in) and the shower mixing valve attaches to them. The valves have adjusting connectors to perfectly line up with the supply pipes.
One huge advantage of these is that replacement of the valve is fully extermal to the wall. These come in thermostatic mixing types.
You could even have the supply pipes emerge from the wall low, turn 90 deg up to place you want the valve then 90 deg out and mount the valve. All this exposed piping could be designed to be attractive.
One company sells "bespoke" shower plumbing valves out of copper pipe mounted externally on the control wall. These, though, might not have a thermostatic mixing valve.
EDIT
The prescribed separation of the two supply pipes that I have seen is 15 cm. You would have to dertemine how close to this the pipes must be located. The external thermostatic mixing valves I have seen have a clever means of adjusting the spacing on the valve connections, which might give as much as 1 cm of adjustment (or more) but that would have to be paid attention to. In the US the normal spacing of the old two valve tub/shower lines was 8 in, so if any plumber is engaged he would have to be informed on what was being used.
