There's more than one way the laser lens moves.
First of all, the lens itself can move up and down (to focus the beam), and laterally (in and out) to track the data spiral on the disc. This is done using a mechanism that lets the lens "float" and be moved precisely and quickly using electromagnets. This mechanism keeps the lens focused on the disc as it spins, and it follows the data spiral as the disc plays, but the range of this mechanism is only enough to play a few seconds of a track before it reaches its limit, so then the other mechanism comes into play, which moves the entire laser assembly in and out, usually a gear driven motor/servo with the entire assembly on rails. This is the noise you hear when you jump from track to track, or when the disc finishes playing, you hear a motor for a few seconds (returning the assembly to its start position at the beginning of the disc).
The former "floating lens" mechanism rarely has friction issues, and if it did, it would probably have trouble playing the entire disc, not just the first few tracks, though the disc does spin faster at the beginning of the disc so the tracking servos do have to work harder. The mechanism that moves the entire assembly in and out is more likely to have binding issues if the lubricant has dried out, which could be causing your issues. While playing a disc, this mechanism has to move in small, precise increments in conjunction with the lens tracking to allow the assembly to stay on track... if you open up a CD player and watch it play a disc, you can see the motor turn periodically as the lens tracks from the inside of the disc outward. When jumping from track to track, this motor runs at a higher speed and probably overcomes the friction better, but if it's sticking, it could have issues with the small incremental turns.
If you can get to the innards, I'd try cleaning the rails with a swab, and verify the assembly is moving freely along its full travel. Some light lubricant might help, but use just a tiny amount and be careful not to get it on the laser assembly or lens, or anything that might come in contact with the CD.
If playing the 1st 3 tracks over and over is helping, another thing to try is to put on the longest CD you have (80 minutes or so), and play the last track to the end, then let the player stop, to make the assembly move over its entire range of travel. Do this repeatedly to spread the lubricant over the rails. Another option is to program the player to play the first and last tracks of an 80 minute CD, then let it run on repeat for a few hours.
