Medicinal is not always caused by infection (and not ever having one doesn't mean you can't get one). That said, it is entirely possible that you are detecting something you associate with medicinal that is coming from the spalt. It's not a flavour I associate with the hop myself but everyone has different palates and palates are informed by psychological factors (memory association etc).
Drinking an alt right now with spalt and tett. I get subtle orange and mandarin flavours (I also get orange with tett) - almost like an orange tea. Very light hint of resin - like chinook toned down by a factor of about 10,000. Very slight floral edge. Tiny hint of woodiness and that tobacco/earth you can get with high amounts of fuggles but again very, very subtle. I'm familiar with the spice descriptor and I get a feeling at the top of the tongue that reminds me of something like all spice but the flavour itself does not. I put it down to the sensation of the resin but I could be pulling myself.
It's why I love noble and noble type hops so much - they are wonderfully subtle, wonderfully delicate and yet massively complex and layered.