Discussions Importance of the Jedi for the Republican war effort in the Clone Wars

Having just finished Star Wars: The New Droid Army, found myself confronted with an interesting question.
If someone haven't seen the game, I'll firstly write a brief summary of its plot - Anakin (the game's events take place almost immediately after AotC, directly dovetailing with it) discovers that Separatists are building new battle droids covered with cortosis which makes them lightsaber-proof. Then the Council orders him to go to Melatorn (where these droids are being made) and destroy the factory what he does, killing Count Dooku in process (the game was released in 2002 and developers didn't know what was Lucas going to do in RotS, so later the Lucasfilm retconned that it was not the real Dooku, but his clone).

Per Yoda, the droid army made of cortosis-covered (and thus lightsaber-proof) droids is a threat that the Jedi (and the Republic since the Jedi then were a part of the Republican army) are unable to handle.

Were the Jedi really that important for the Republican war effort? In AotC, Mace Windu states clearly that there isn't enough Jedi to protect the Republic. We don't know the Jedi Order's size as of 22 BBY (at least in Legends) so it's impossible to calculate the Jedi-clone ratio in the GAR. And let's not forget that after the Order 66 the Empire managed to take out the CIS remnants without the Jedi. So how really important were the Jedi for the GAR?

Since the emergence of the lightsaber-proof battle droids doesn't affect the Jedi's ability to command the clone armies (in fact, Jedi usefulness as generals is also very doubtful but let's assume they are competent enough), I want this discussion to focus particularly on importance of the Jedi in combat (using their lightsabers and the Force), not their role as commanders,

And yes, I know that the whole Clone Wars were a farce aimed at killing the Jedi. But Yoda obviously didn't know that before RotS' events took place so his appraisal of the Jedi's role in the war is based on premise that it's a real war.

 

I'd say yes because...even if it doesn't make sense, the plot says so.

I took this somewhat analytical approach to the clone army, and no matter what, the idea that one small water-covered planet growing a few thousand clone every 10 years or so would never come close to the number of droids the Separatists could produce. The Separatists could probably build a droid in a few days, so you're talking several thousand battle droids per clone trooper.

But somehow the clones can fight off the droid army, because the plot says so.

 

The Republic Commando books gave the numbers of three millions of clones and several hundreds of millions of droids. So in fact there are from one to three hundreds droids per clone trooper.

And droids are not only much faster in production, but also much cheaper. And it was possible for the CIS, if necessary, to double the droid production very quickly, which the Republic obviously couldn't do with clone production.

A possible in-universe explanation could be that Dooku actually didn't want the CIS to win and sabotaged the CIS war effort (as he did on Kamino where he gave the Separatist forces wrong information).

Otherwise the Republic would be screwed.

 

I would say that the Jedi (when acting as generals) actually hurt the Republic war effort - since they don't have the military and officer training to know what they're doing. They're armatures, and command of the Republic armies should have been left to military officers who actually know their business. That the Jedi tended to lead from the front rather then their HQ made it worse.

 

Well I mean the point of the war was to kill jedi, so while they are likely very important, remember that the goal of the war is to kill Jedi.

 

For me the Jedi Droid Army video game is a holoshow