Its certainly not a given that we will lose the EHIC scheme or new equivelent (If we continue to provide similar for EU citizens in UK) , or higher insurance premiums, or potentially having to pay internet roaming fees, or lower compensation on any flights, or longer processing times at passport control (for tourists) etc etc.
My point is that in reality its best for both sides if little changes post Brexit and it my guess that's exactly what will happen. We will CONTINUE to show our passports whether Blue or Red and we will continue to travel freely through the Schengen area. The rest is just speculation EITHER WAY.