I’ve just completed my first attempt at putting a natural finish on a pair of Baltic birch baffles. I put three coats of Boiled Linseed oil on them, and it brought out the grain pretty well. I know this is not a wood known for a nice grain appearance.  I’d kind of like to have a slightly glossy look. Can I put on a top coat of semi gloss polyurethane? Would this help or hurt the final appearance? I also wondered if it might help show the grain a bit more?

I once finished some Baltic birch with Danish oil  and then wax.
I seem to remember spending a lot of time buffing it down smooth and then waxing it about eleventy seven times. It was real smooth.
You might not want to go as psycho as I have been known to do.

Try a nice beeswax.   If it doesn’t give you the results you want, it will be a hell of a lot easier to take off than trying to undo a polycoat.  Plus with a beeswax you don’t have to be as concerned with humidity when applying.

Dominick

I've waxed over stain but never linseed oil.  The difference is that linseed oil takes more time to dry.  I'd just make sure you've given enough cure time.  Steve may have insight on this since he's waxed over LO.

I've always used Minwax wax which is super easy.  Wipe on, wait a few minutes till dry and buff off with a rag.  The only time I got in trouble was with a commercial floor wax that needed a powered floor buffer with boar's hair bristles to cut it.

You will get a nice sheen -- somewhere between semi gloss and gloss.  The wax probably won't darken much since it's such a thin coat.  The dark wax is more so it won't contrast with light wood in nooks and crannies.

Come to think of it, I even waxed over the black paint (after first sanding) on my Big Betsy baffles.

I find that one coat does it for me.  Depending on how you put on a second coat, the wet wax could melt the first coat and you'd be basically back to one coat.

I have used minwax wipe on poly over stain on baltic birch ply several times.  
https://www.minwax.com/wood-products/clear-protective-finishes/wipe-ons/minwax-w...

It never seemed to get quite to the finish I envisioned, fast enough(# of coats) to suit me.  
Used these between coats.
https://www.amazon.com/3M-10144-Between-Coats-Finishing/dp/B00004Z480

So then I sprayed some urethane over the top of the wipe-on after it was good and dry.. and that did it.  
https://www.menards.com/main/paint/interior-paint-stain/polyurethane-clear-prote...  <-- this comes in satin, semi-gloss and gloss - pick your amount of shine...

And a little wax to top off.. and it looks really nice.

* just remember you can put water based over oil, but not oil over water based.. so pay attention to which version you buy - most come in oil and water based formulations.  I like to start with oil... top with water based.