Been giving this idea some thought ...
I'd plane all the boards so they were exactly the same size and glue up your strips so you end up with "boards" 3" thick, 3" wide and however long the width of your desk top will be (commonly around 30"). Then I'd run all 4 long sides through a joiner to get smooth surfaces. Then I'd cross cut those into 3/8" thick slices - you end up with pieces 3" x 3" x 3/8", with the end grain showing on the 3" x 3" faces. Ten of these then get glued together into 3" x 30" x 3/8" boards. Then run the 30" x 3/8" faces through the joiner to get smooth faces again and then start gluing those to your birch plywood underlay. Alternately, you could glue 2 or 3 of your end-grain boards together and then join the outer edges to smooth them and then glue those together on your birch plywood underlay. Then you're going to have quite a bit of planing and sanding to smooth up the end grain surface. I'd sand it down to 150 grit, maybe 220 max at this step.
Then, here's what I'd do. Mix up some two part epoxy, and using a scraper or piece of bake-a-lite or some flat edge I'd squeegee the epoxy onto the end grain surface, trying to get as much epoxy into the softer areas of the end grain and filling any gaps left from the seqmented glue ups. You want to leave a very thin layer of epoxy over the entire surface. I used this method on my last guitar to pore fill the surface before final sanding. After the epoxy dries, you do your final sanding, trying not to sand away all the epoxy, just get it flat and smooth. If you want to stain the end grain surface, do that before you do the epoxy fill. I think if you do it this way, you'll minimize the potential for soft parts of end grain, and achieve a very smooth flat, durable writing surface that still looks like wood (i.e. doesn't look like it has a thick epoxy coating). This video explains and shows how a luthier uses this epoxy fill method to achieve a very smooth surface.
After the top is cured, carefully sand it down to 220. I'd plan on gluing on mitered edge strips to finish the edges, maybe use a contrasting wood. Then do your final sanding and apply your finish coat wax or whatever.
