You need to re-examine your statement that “3. Checking/splitting of solid timber - I’m ok with this, part of having natural wood”.

Read the paragraph entitled “Shear Failures in Actual Structures” (page 2) of the article at the link: https://www.fpl.fs.fed.us/documnts/pdf1996/ramme96a.pdf

If you don’t like the look of stained glulam, you can always cover them with the lumber of your choice.

The trick is to avoid long spans and large point loads for heavy timber beams.

Ask your architect.

Ask your architect or builder to give you pricing alternatives.\

The exterior beams, whatever they are, will need protection from the elements, particularly the beam ends and tops.

Here is a reference about protecting exposed heavy timber

Glulams may not stain well; however, if you're painting the exposed beams, that shouldn't be an issue.

Something you might discuss w/your architect and/or builder if you want to stain the beams in your great room is how practical it might be to use actual beams in the great room only but use glulams in the other rooms where the beams will be exposed and paint those beams.

Something else you might consider is using glulams throughout, but in the great room, have the glulams wrapped in actual boards and stain the boards.

Let us know how your project goes.