*deep breath* Okay. So most of the story proceeds as the book. There are minor differences with who does and says what and what their motivations are. I will cover the major changes:
Things went roughly the same until they got to
Moria.
Glorfindel was the leader, in lieu of
Gandalf. While in Moria Glorfindel caught
Gollum while they were camped for the night. Gollum was tied up in Elven rope and Glorfindel was unsympathetic to his whining about how it burned. This proved less helpful when the constant complaining alerted the
Orcs to their presence. Glorfindel confronted the
Balrog on the Bridge but had his spear destroyed and was thrown to the far side.
Radagast then sacrificed himself by destroying the bridge, with the Balrog on it, and falling into the pit.
They proceed to
Lothlorien. Gollum's month long stay in Lothlorien is transformative and
Frodo decides to take him with them. Here, again, much of the story proceeds as the original until Rauros. Frodo, having contemplated the effect
the Ring had on Gollum, decides to proceed alone but is, of course, caught by
Sam. Gollum rightly points out that they need his guidance and he goes as well.
Merry and
Pippin are taken by
Uruk-Hai but the rest of the Fellowship remains in tact and no one dies (Glorfindel is
the one ambushed but he'd been given a reforged Aiglos by
Galadriel to replace his destroyed spear).
The Fellowship track the Uruk-Hai across the
plains. Meanwhile Frodo, Sam, & Gollum make much better time through
Emyn Muil and are two whole days ahead of the original story. In
Fangorn,
Aragorn, Odin, Glorfindel,
Faramir, & Folco encounter Radagast the Green. It seems he fair slightly BETTER than Gandalf once he and the Balrog hit the underground lake as Radagast shape-shifted into a giant squid (he's a master of "shapes and hues" after all) and succeeding in keeping the Balrog extinguished in the water until their fight drew the attention of the Nameless Things that dwelt in the water. They attacked them both but Radagast shape-shifted into a small fish, escaped their grasp and notice, and made it to the shore. He waited for the Balrog who slew the
creatures and then headed up the Endless Stair. Radagast followed and they emerged at the peak. The Balrog was exhausted from the constant fighting and climbing but Radagast was comparatively fresh. The Balrog was unable to reignite until late into the fight and so Radagast prevailed but was also slain (he was still much less powerful, or martial, than the Balrog). Like Gandalf he was sent back to aid in the fight.
Here things return to the main narrative. The healing of
Théoden, of course, goes differently. Things really change once we get to
Orthanc. Radagast's pleas hit differently than Gandalf's. They were sent from
Valinor together, after all (at
Yavanna's pleading), and
Saruman was never jealous of Radagast. Saruman relents and
places himself into Radagast's custody and hands over his
Palantír. There's more to this than I'm going into but I don't have the space to lay it all out. Pippin looks into the Palantír and he, Radagast, and Saruman go to
Minas Tirith.
Meanwhile, Frodo, Sam, and Gollum happen upon
Rangers in
Ithilien.
Boromir (who had been out gaining news on the
Ranger's status) wishes to take them back to Minas Tirith when he perceives they carry something of importance to the Enemy but Gollum assaults him and Frodo, Sam, and Gollum escape. They
travel through
Cirith Ungol safely and are uncaptured.
Pippin in Minas Tirith goes largely the same as the book save
Denethor and Saruman verbally spar. Boromir is there but keeps silent about finding Frodo, Sam, and Gollum as he is embarrassed that they escaped but Saruman and Denethor perceive he is keeping information hidden.
Folco is taken in by
the Grey Company (specifically
Halbarad).
The Battle of Pelannor
Fields proceeds largely as the book. Halbarad is saved by Folco. Saruman is the one who helps Pippin save the unconscious Boromir from Denethor's madness (Radagast was unable to keep Saruman in tow during the battle so he'd placed him under "house arrest"; Pippin went to Saruman first as Radagast had transformed into a
Great Eagle and was engaging the
Nazgûl; Saruman was not motivated by goodness but by a desire to regain his pride (and Denethor was acting like a fool)).
The march to Morannan is largely the same save there is no fight; the Ring is destroyed before the
Mouth of Sauron can even speak.
In
Mordor Frodo, Sam, and Gollum evade all detection and make it to Mount Doom. When Frodo refuses to destroy the Ring, Gollum is incensed at the betrayal of his master and attacks Frodo. They fight, Gollum bites off his finger, and then he falls into the lava. Frodo and Sam are rescued by Radagast in the form of a Great Eagle.
Things proceed as the book again until Gandalf arrives for the Coronation (he'd been up at
Erebor during all of this, where his efforts saved many lives). Gandalf is surprised at Radagast's growth but before he can speak to Saruman, Radagast intervenes and states there's nothing more to be said: Saruman will return to Valinor as Radagast's detainee. After the coronation everyone returns home. Radagast sets sail immediately while Gandalf wraps up any loose ends. Boromir abdicates the role of steward to his brother (in shame of failing into temptation compared to his more valourous brother). His is assigned to
Osgiliath and retains his role as commander of the
armies. Gandalf, Glorfindel, Galadriel,
Elrond, Frodo, &
Bilbo set sail across the sea.
To my answer in which I basically said, no, it would not be more reasonable, the person replied that the plan in the book is "stupid"; that 
