Strictly RAW: Your DMs are correct. Thrown weapons do not have ammunition.
Personally, I'd consider whether multiple thrown weapons can be used in situations such as this or Horde Breaker to be strictly in the DM's prerogative. It makes a fair amount of sense that a character could be good enough with throwing knives to put multiple in the air in 6 seconds. On the other hand, it's pretty crazy to think that someone can throw multiple spears in 6 seconds, considering the windup you'd need to throw them.
Another "RAW" point against it is that drawing a weapon is a free object action. Despite it being "free" you only get one a turn. So even if you could throw multiple knives or spears, you're violating the free object action rule by drawing more than one.
In conclusion, your DMs make the rules of your game, and they've ruled that thrown weapons cannot be used in the way you're suggesting. Nothing in the rules disagrees with them. The best course of action is to talk to the DMs and explain why you want to throw multiple weapons, and listen to their reasons for disagreeing.