He could have simply overwhelmed the Jedi Knights with his stormtrooper army and be done with it.
The problem with simply taking control of the Republic by force and wiping out the Jedi is that it casts Palpatine/Sidious as the villainous military overlord ruling by force (no pun intended). If he moves too quickly and uses too much force too early, he risks planets and systems rebelling.
The more you tighten your grip, Tarkin, the more star systems will slip through your fingers.
Instead, he chooses to play the long game.
He creates an enemy for the entire Republic to rally against - the Trade Federation and their droid army. As the war breaks out, he tricks the senate into granting him emergency powers - powers he never intends to relinquish.
From Attack of the Clones:
MAS AMEDA This is a crisis! If the Senate votes the Chancellor emergency powers, he could approve the use of the army in a minute.
PALPATINE Please, please, I don't wish to have emergency powers. That's too extreme a solution. It's akin to a dictatorship. We must rely on the Jedi. Master YODA, how many are available to go to Geonosis?
MACE WINDU and YODA look at one another.
YODA Two Hundred,... less or more.
BAIL ORGANA With all due respect for the Jedi Order, two hundred will be no match for hundreds of thousands of battle droids.
MACE WINDU Patience. We should wait for Obi-Wan to report back. We don't know that Count Dooku has made a treaty with the Corporate Alliance, it's speculation.
SENATOR ASK AAK But we must prepare for the worst. I'm going to propose a motion granting emergency powers to the Chancellor at the next session. We must not wait!
PALPATINE Out of the question, Senator! You and I are too closely aligned. The issue will become partisan and debates will begin. The proposal must come from a neutral source.
MAS AMEDA If only Senator Amidala were here.
JAR JAR steps forward from the back of the group.
JAR JAR Supreme Chancellor... my august colleagues, I would be proud to propose the motion in question. This is a grave situation, and I'm sure Senator Amidala, and the Queen of Naboo would agree.
SENATOR ASK AAK Thank you, Representative Binks.
Silence. Then PALPATINE sighs deeply.
PALPATINE If called upon, I will serve. But it will be the saddest day of my life.
Now, by letting the war drag on and appearing as a steadfast leader, he ingratiates himself with the people, becoming more and more popular. Some planets/systems/territories become more reliant on the Republic as they become war-torn. The Jedi, meanwhile, dwindle in number as some of them are lost in battle.
Over time, Palpatine uses the closed off Jedi Counci. against themselves and officially takes oversight of the Council.
From Revenge of the Sith:
ANAKIN stands with PALPATINE at his window overlooking the vastness of Coruscant. Several buildings have been destroyed. A brown haze hangs over the landscape.
PALPATINE: ANAKIN, this afternoon the Senate is going to call on me to take direct control of the Jedi Council.
ANAKIN: The Jedi will no longer report to the Senate?
PALPATINE: They will report to me . . . personally. The Senate is too unfocused to conduct a war. This will bring a quick end to things.
ANAKIN: I agree, but the Jedi Council may not see it that way.
PALPATINE: There are times when we must all endure adjustments to the constitution in the name of security.
ANAKIN: With all due respect, sir, the Council is in no mood for more constitutional amendments.
PALPATINE: Thank you, my friend, but in this case I have no choice . . . this war must be won.
ANAKIN: Everyone will agree on that.
PALPATINE: ANAKIN, I've known you since you were a small boy. I have advised you over the years when I could ... I am very proud of your accomplishments. You have won many battles the Jedi Council thought were lost . . . and you saved my life. I hope you trust me, ANAKIN.
ANAKIN: Of course.
PALPATINE: I need your help, son.
ANAKIN: What do you mean?
PALPATINE: I fear the Jedi. The Council keeps pushing for more control. They're shrouded in secrecy and obsessed with maintaining their autonomy . . . ideals. I find simply incomprehensible in a democracy.
ANAKIN: I can assure you that the Jedi are dedicated to the values of the Republic, sir.
PALPATINE: Nevertheless, their actions will speak more loudly than their words. I'm depending on you.
ANAKIN: For what? I don't understand.
PALPATINE: To be the eyes, ears, and voice of the Republic . . .
ANAKIN thinks about this.
PALPATINE: (continuing) ANAKIN . . . I'm appointing you to be my personal representative on the Jedi Council.
With his emergency powers, control of the Jedi Council and effective power over all of the senate, he can now legally assume control over everything.
BAIL ORGANA: Now that he has control of the Jedi Council, the Chancellor has appointed Governors to oversee all star systems in the Republic.
FANG ZAR: When did this happen?
BAIL ORGANA: The decree was posted this morning.
PADME: Do you think he will dismantle the Senate?
MON MOTHMA: Why bother? As a practical matter, the Senate no longer exists.
GIDDEAN DANU: The constitution is in shreds. Amendment after amendment . . . executive directives, sometimes a dozen in one day.
Eventually, he has amassed so much power, and the Jedi have become so weakened that they can't stand a chance against his betrayal and creation of the Empire. The end of the war also provides a good point in time to frame the Jedi as rebels. As the war ends, the Jedi come to apprehend Palpatine for being a Sith Lord and conspiring against the Republic. But without the public knowing any of this, their failed attempt can easily be spun as a failed assassination in a plot by the Jedi to take control of the government.
From Revenge of the Sith:
MACE WINDU: I sense a plot to destroy the Jedi. The dark side of the Force surrounds the Chancellor.
Kl-ADI-MUNDI: If he does not give up his emergency powers after the destruction of Grievous, then he should be removed from office.
MACE WINDU: That could be a dangerous move ... the Jedi Council would have to take control of the Senate in order to secure a peaceful transition . . .
Kl-ADI-MUNDI: . . . and replace the Congress with Senators who are not filled with greed and corruption.
YODA: To a dark place this line of thought will carry us. Hmmmmm. . . . great care we must take.
ANAKIN: Master Windu, I must talk to you.
MACE WINDU: What is it, Skywalker? We are in a hurry. We have just received word that Obi-Wan has destroyed General Grievous. We are on our way to make sure the Chancellor returns emergency powers back to the Senate.
ANAKIN: He won't give up his power. I've just learned a terrible truth. I think Chancellor Palpatine is a Sith Lord.
MACE WINDU: A Sith Lord?
ANAKIN: Yes. The one we have been looking for.
MACE WINDU: How do you know this?
ANAKIN: He knows the ways of the Force. He has been trained to use the dark side.
MACE WINDU: Are you sure?
MACE WINDU: Then our worst fears have been realized. We must move quickly if the Jedi Order is to survive.
Palpatine wants power and he knows that the way to get it is through the war.
Sidious needed the war to convince the Senate to give him control, first. Once he was named Emperor, he had the power to dissolve the war and attack the Jedi, but to do so would mean exposing his treachery. Instead, he was able to use the fake war to frame the Jedi as traitors and secure his position as Emperor with the general population believing he is still good.
In other words, if he had simply overwhelmed the Jedi with the army he had made:
- He would not be able to become Emperor, as there would be no perceived threat of the war with the Droid Army
- Sidious would be hated for killing the still-liked Jedi, or if Palpatine took control of the Army and killed the Jedi, he would have been deposed as chancellor.
- He wouldn't be able to pay for the Clone Army, as he wouldn't be the leader of the government as Sidious, or if he did it as Palpatine he would not be Emperor and therefore would have to pay for the army through the Senate who would have already deposed him for betraying the Jedi.