@themanj Tanks work poorly in cities, even mudbrick ones like this one, since they can get caught under buildings or over piles of rubble, and in this town the streets might well be too narrow to let them through (and catching two walls at the same time can also stop the tank, or make it throw a track). Given they prefer zero own casualties (or at least provable such, for propaganda reasons if no others) they stick to letting specialties work where they are most suited.
As for the wooden gate, a HEAT is so much overkill that it might not work particularly well at all. It's so very directional. (It doesn't fight the structure - it's task is to make a small hole and then expand and kill the people behind the protection.)
On the other hand, a tank crew would know that as well, and fire a HE round (which is what you use to fight light fortifications) so that's just a mangaka error. (Especially given that the result images also look like HE effect.)