You can definitely play the way you're doing it, but I personally prefer to keep a team of relatively equal level so I can counter nearly every Pokemon I come across. I've always played this way; I'm too scared of blacking out. I like my money! ;)
As you've recognized, if your Charmander blacks out you're in trouble :P If you continue through the game the way you are playing you might encounter some difficulties (namely, Misty), as already pointed out in wonderful detail. Also consider that you will have to battle your rival on multiple occasions, and he will always have the hard counter to your Charmander. On top of a Kadabra, soon enough. Ugh.
Another thing to worry about is the level restriction. If you fight with primarily one Pokemon, it is going to gain a lot of experience through the many trainers you cannot avoid or accidentally walk in front of. But if it gets too high-levelled, it will not obey you until you have the proper badges. And then you're REALLY screwed :P You're capped at level 30 until you beat Erika, who is quite a ways away, so be careful :) After Erika you're pretty much okay on the level problem; you get enough room that you probably won't hit the cap.
Oh, another thing. A lot of early Pokemon aren't great, it is true, but I highly, highly, highly recommend catching an Abra and levelling it up (you have to send it out first and then switch immediately to another Pokemon since Abra can't actually attack--don't worry, Abra doesn't need many exp points to level and evolves at 16) to evolve into Kadabra. Kadabra is an amazing Pokemon to have. Weak in defence, but with incredibly powerful attacks and fast enough to hit first.
Because its evolved form is so good, Abra are hard to catch--their only move is Teleport, so as soon as you find one it just Teleports away. I usually level up a Pikachu along the way (I need Pikachu anyway for Flash in Rock Tunnel) until it's fast enough to Thunderwave an Abra before it can Teleport away. Then I pray it remains fully paralyzed so I can toss out a Poke Ball. It can take what feels like forever because often they manage to Teleport regardless, but it is worth it. A better way to catch an Abra is to put it to sleep, but it's a lot more annoying to level up a Jigglypuff or a Butterfree. Pikachu level up easily because they are super effective against the Zubat in Mt. Moon and the Pidgey which are ... well, everywhere. Pikachu are also quicker than the other options, which is good since Abra are pretty fast.
You can first find Abra in the grass if you travel the Nugget Bridge north of Cerulean City and then go left a little, the long patch of grass below has a 15% chance of yielding Abra. If I pick one a lv 10 one then and switch it in and out while fighting the trainers on the way to Bill's house, it without fail evolves by the time I've gotten through them :D
Also! (I talk too much, I know.) People tend to think Rattata is a bad Pokemon because it is so weak. However, it's fast enough that you can level easily against early trainers and in Viridian Forest and it learns Hyper Fang at lv 14 which is a super good attack - especially because it learns it so soon. It'll take out Pokemon around its level easily with Hyper Fang. And when Rattata evolves (lv 20, so it'll be doing that early; mine always evolves somewhere in Mt. Moon or soon after) its attack and defence jumps quite a bit higher. Suddenly it's not so weak ;) My Raticate is actually my strongest Pokemon, going by stats, even stronger than my starter Squirtle/Wartortle (though unfortunately my Squirtle happens to be a less powerful one this time around); and it's second highest to Wartortle in defence. And because Raticate's type is Normal, it doesn't have a weakness. It can't attack Ghost Pokemon, of course, and it does very little against the Ground and Fighting types, but the awesome thing is you can teach it an attack that will help it counter those types with ease. (I like teaching it Dig, and later sometimes Ice Beam if I don't feel like giving that to Blastoise.)
Okay, now I'm done. Sorry, I love Raticate. I keep it with me the entire game, and use it to help take down the Elite Four. I end up with my starter, a couple of Legendary birds (one with Fly), a Kadabra, another Pokemon which varies (sometimes a super low-level Grass Pokemon so I can cut all the stupid bushes--I don't like fighting with Grass Pokemon so I just give one Cut but never use it), and then Raticate. It's such an underrated Pokemon.