I'm just wondering if I one should see the piano like a percussion instrument? Where the left hand is on some beat that develops. Basically, if you take the view that the piano is a percussion instrument, should you play it more like a drum? In some ways it resembles drum pads that you strike with your fingers.
As with most things musical, there's nothing to stop you approaching piano in this way.
Let's talk about percussion.
There are many percussion instruments that are not drums. Glockenspiels, marimbas, vibraphonii; the list goes on. They do have some percussive aspect to the sound, so you can use them rhythmically. But they also have strong melodic and (in some cases) harmonic potential.
The piano is even more versatile. It can be strongly rhythmic. It can be sweet and melodic. It can lay the harmonic foundation, or it can dance around on top of it. It can do all of those at the same time.
In short, the sound that is produced is far more important than the mechanism for producing the sound. As a result, rhythm is a possibility, but far from the only way to approach the piano.
Now that's out of the way, we can think about how we can treat the piano as percussion.
I have to do this on occasion: I play in groups with just a few singers and a piano. If we're playing certain songs, the piano takes on a very rhythmic-harmonic role.
In its simplest form, you can treat the left hand playing bass notes as a bass drum, and use your right hand as a snare (playing block chords).
Branching out a little, you can play arpeggios or other repeated figures. I sometimes do this up high, and it works a little like a hi-hat. Another example is that eight-on-the-floor thing piano players do in some rock songs. Just hammering out quaver chords up high all through the chorus. It's quite effective.
Drum kits are a good source of inspiration for rhythm. But don't expect the piano to fill exactly the same role. It can be considered percussion, but it's not a drum.
Your basic premise is skewed. You've heard that the piano is a percussion instrument. It can be classed as such, because the sound is produced by hitting. Hitting strings with hammers that are activated by pressing keys.
You've heard that percussion instruments are mainly drums and such like, all of which can be hit to produce sound.
You've put two and two together - and found an answer. But not the whole story. Four is also the answer to lots of other questions, not all related ! (What's the French for oven?)
Listen to a bass guitar; a tuba; a Sousaphone, basically any instrument can produce what you propose the l.h. on a piano might do. Rhythm. Your misapprehension that just because a piano might be a percussion instrument equates to 'it must/can make a rhythm' is flawed. Of course it can - just like most other instruments. A rhythm is a set pattern repeated. Your mouth can do that. So mouth is a percussion instrument??
Let's take 'theory' of piano further. It has strings. Ah, yes, strings produce beautiful melodies - just listen to a well played violin. So, the piano is now a melodic instrument. True, it does that as well. It's already well documented in previous answers that some percussion instruments play melodies well.
But wait. It's also a piece of furniture! Put a vase, pint of beer (please don't!) on it. Yes, it's that as well...
Trouble with not understanding a full picture is that any theory thought up will be flawed. Or are you just playing Devil's advocate..?
The percussive action in a piano, where something actually strikes the vibrating medium, happens when the hammer inside the piano strikes the string. When played normally, you don't have direct control over this; you can't control the striking position, or angle, or what you're hitting it with; you also have limited influence on how the damping works. So when you play single key on a piano it's rather like playing a very sanitised version of a percussion instrument. You can play rhythms on one note by controlling velocity well, but you'll probably struggle to make very interesting music unless you also introduce some harmonic/melodic interest. This is one way you could see styles like stride and boogie-woogie piano - they are somewhat rhythmic and percussive, but also add that element of extra interest on top.
Another way you could get more in touch with the piano as a percussion instrument would be to get inside it, work out how to undamp the strings, and play it like a hammered dulcimer. But I'm guessing there might be risks for you and the piano - just getting hold of a dulcimer might be more rewarding!
In the art music realm the works of Bartok & Cage are illustrative of piano as percussion & piano as a struck string melodic instrument. (There are many other composers whom you could use as models for either, but these two are easy exemplars to use.)
Look at Bartok's Dance Suite, Sonata for two pianos & percussion, or Music for strings, percussion & celeste for examples of traditional keyboard techniques married with purely percussive exploitation of the instrument. Expanding on the suggestion from @topo_morto to look inside at the instrument's medium, have a listen to Cage's Sonatas & Interludes: modifying the instrument itself can lead to novel effects that you can exploit as percussion & as a melodic instrument. (Just be careful, some modifications -- "preparations" in the jargon -- are destructive!)
I think the view the "left hand is on some beat that develops" is a bit limiting in that the right hand can play just as much a role in the establishment of the beat as the left; conversely the left hand can avoid the beat as much as the right hand. I understand that your question comes from a non-art-music point of view, yet all traditions can teach all others.
One last thing, in traditions coming out of jazz the piano is a member of the rhythm section (as you know): it's worth thinking about the percussive capabilities of the bass & the melodic potential of the drums in this light.
When introducing myself as a percussionist, I get mildly offended when people respond with "so you play the drums right?" The spectrum of percussion instruments spans far and wide; most pertinently, percussion instruments aren't just all about the rhythm. The marimba is one such example, it does not necessarily provide the beat in the same way as a drum set does. Playing the drum set and playing the marimba requires somewhat different sets of skills.
One could argue that within the spectrum of percussion instruments, the piano more closely resembles the marimba than the drum set. Both are keyboard instruments, and both are capable of playing counterpoint (2 or more parts at once). Both are capable of producing steady rhythmic beats like the drums, which would be suitable if you're playing a march for instance. However, they can also play melodies that flow like a river - Debussy's compositions come to mind here. It all depends on the piece you're playing and the directions given by the composer!
There are lots of different reasons to want to classify instruments. It sounds like you're classifying based on function within an ensemble. In a jazz big band, the piano is indeed part of the rhythm section (along with drums, bass, and guitar), a group responsible for laying down the basic structure of the piece. Piano can serve a similar function in many other types of ensembles as well.
Another reason to classify instruments is to understand their capabilities. The piano makes sound via a hammer striking strings, the result being that it can't modify the sound after starting it (other than to stop it), and the volume will decrease over time. This is similar to most percussion instruments... but also to harp, guitar, and other non-bowed string instruments. So it's a similarity but not a uniquely percussion-like one.
I saw a fella playing all sorts on thepiano once. He did "Satisfaction" (R. Stones) and when it got to the couple of bars of the stompy snare, he grabbed the piano lid and bashed it down to make the noise. Got everyone's attention :-D
Piano is an instrument. If you bash the top with your hand, it's a percussion instrument. If you play 4 notes all one semitone apart, you're probably not goigng for something melodic. If you play sweet melodic floaty chords, maybe it's less percissive. If you ... etc.
I think the word "should" is the problem here.
You can do what you want! If you find a way of playing (anything) that results in something unique or unusual, and you like it, then go with it!
You invented something. :-)