Wondering if my Tiagra shifters are fixable?
Fixable?  Possibly - but there will be risks, and you need to decide if the risks are worth accepting.

Downsides:


Its your brake lever - and there's a 50% chance its your front brake lever.  The front brake is the most effective brake on our bikes, because it does 90+% of the braking.  Any bending of the lever back into line will stress the metal underneath.  I can't tell what model of shimano, but there's a high chance its an aluminium (aluminum) bar and they take poorly to being flexed.
The nameplate is also a rain cover, keeping moisture and dirt out of the mechanism.  Without that, the shifting mech will get worse over time as dirt collects in the clicking parts.


Personally?  I'd keep riding it as-is (gently!) without attempting to straighten it, and I'd also start searching for a suitable replacement lever on ebay or whatever your local sources of parts are.  This way you can still ride it for now, though I would not recommend long rides or any racing or back-country rides.

If the "feel" changes, if shifting gets bad quickly, or if anything goes wrong, stop and re-evaluate right there.

If the lever broke while being straightened, then you can't ride it till a replacement is found and fitted, and there's less chance of finding a cheap/matching one - you might have to buy new which can be expensive, and you might end up with a mismatched pair of levers, which is super-irritating.

Answer Replace the lever with a known good one.  Used is fine.