My guess is “push a given hero” refers to a classic hero provided in TC. “Is it better to broadcast?” refers probably to summoning 
I won’t answer this, because this may go wrong either way. Summoning may provide much better heroes, of course, but yet again, there are so many complaints like “did 60 pulls or 100 pulls and I got only garbage”.
Anyway, the emblems should be used. Like you already said, the healers should take the HP/def path. I will continue with this:
- Support heroes should take HP/def path
- Rogue class should take ONLY atk path
- Monk class is very resistant to all kind of status ailments and most heroes here should also take the atk path
- Paladin class should take the HP/def path excepting the snipers. This particular path is poor in atk nodes and no matter what, the snipers are forced to take also def path. So, for a hero like Sonya, for example, should be chosen any atk node she can get. Her def stat is already impressive, but being a sniper should also do more damage.
- Wizard class is trickier than any other class. Its ability requires naturally a better offensive, but the most heroes in this class are poorly chosen. Why anyone would increase the attack for Guin, Kiril or Proteus? Not even for Isarnia, because she’s squishy and unfortunately the most atk nodes within wizard path are combined with HP nodes, not def.
- The fighter class suggests is better to choose atk path. If the fighter dies, revives and fires, at least to do some good damage. The def path doesn’t help a fighter on reviving, obviously.
- Cleric class is also a defensive class. Excepting Elk, who should take only atk path. Actually his skill relies on this.
- Ranger class is the one within the heroes are rangers indeed. And even the name suggests the atk path should be the chosen one. One shot, one kill. At least, this is ability about. If bypasses the defensive buffs.
- Sorcerers, druids and barbarians should be chosen based on the first “rule”. Healers and support heroes with HP/def node.
But in the end, any player should make the own choices. The player knows better where and when uses the heroes.