There's plenty of scenes in the LOTR movies that don't employ CGI. Most of the battle scenes have hundreds of costumed extras (in some case thousands of them). The film is also notable for its use of models to represent places such as Helm's Deep and Minas Tirith, even though CGI could have been used. Sometimes the models are CG enhanced, of course, with little added visual elements inserted into (or onto) the models.
Obviously there are sequences that are CG - the scenes with the Oliphaunts, for example, would have been impossible any other way. Some of the scenes with horses would have been extremely difficult to safely do with real horses. Clearly much of the stuff with "Legolas - Super Ninja" was also CG, but on the whole the LOTR films are pretty much the last major fantasy movies that employ extensive modelwork, large armies of extras, and scenes shot on location rather than in front of a greenscreen.
I must say I missed the scene of "600,000 Rohirrim" riding to Minas Tirith! Although Peter Jackson definitely increased the size of the besieging forces to around 250,000, I can't say that I thought the Rohirrim were exaggerated in numbers. As I recall those scenes were shot with about 300 real riders and horses. Extra riders were added in using CGI.
The reality is that even in the "old days" the number of people in a battle scene was often exaggerated by clever use of camera angles. Films like "Lawrence of Arabia" and the Russian version of "War and Peace", where there really were thousand of extras, were the exception rather than the rule.
If LOTR was filmed without CGI then it would have been more or less impossible to do. Stanley Kubrick contemplated doing it in the late sixties after
2001, and even he thought it was impossible to film at that time.
There's also the problem of actually costuming the extras. It's one thing to have thousands of extras but it's another thing entirely to have to put 20,000 people in uniforms! 20,000
orcs is even more difficult. Stanley Kubrick (yes, him again) developed a method of mass producing military uniforms for his version of Napoleon (which never made it past pre-production) by printing the uniforms onto a special kind of paper. So you would have seen armies of thousands wearing
paper uniforms if that film had ever been made! Kubrick also planned on shooting much of the film in the studio rather than on location. If you've ever seen
2001 then you may be aware that the entire film (except for one shot) was filmed entirely inside a studio, even though the first 20 minutes are set in Africa. Kubrick pioneered a revolutionary technique where huge high-definition images could be projected at an angle onto a special reflective screen placed behind the actors, this resulted in a far more realistic effect than the more commonly used "rear-projection" technique. It also allowed far larger images to be used - you could in effect depict an entire huge landscape, something that wasn't possible with rear projection. If Kubrick had made LOTR then he would have undoubtedly made use of this technique.
The real problem with the LOTR films isn't excessive use of CGI, it's more to do with a failure of tone. There are moments where Peter Jackson seems to be utterly tone-deaf in terms of the material. However, this is not unusual in film adaptions - imagine if you will a
Cecil B. de Mille version of LOTR. Sure he might have used thousands of extras, but he would have also dumbed-down the material dreadfully. Even worse, in modern terms imagine a
Michael Bay version of LOTR!
You'd be amazed by the number of films that use CG imagery these days. It isn't just fantasy and sci-fi. Almost all historical films use it now - if you're shooting a movie set in the 1920s it's a lot easier to just digitally remove any inconvenient modern elements from a shot rather than having to try and "shoot around" things.