If the whole ECU is bad, then this is true. However just one small circuit can go bad in an ECU, causing one small item on the vehicle not to work, while everything else works fine. I had this happen on my Infiniti G20. Car ran great, but check engine light came on. EGR code. Tested out everything and finally traced the problem back to the ECU. Turns out the ECU was getting all the right inputs, but when it was supposed to provide a ground to operate the EGR, it wouldn't. All grounds were good, just something internal in the ECU. Another ECU solved the problem instantly, as I watched it respond on a multi-meter.

If the cold start injector has not been working for some time (whatever the cause), then it is possible for it to get clogged up from non-usage. It might be worth taking a look at, since even if you solve your problem somewhere else, the injector may still not work.

Just my experience, but a clogged fuel filter usually causes your engine to sputter when it's under a load. Although I wouldn't limit it to just that, that is usually where the problem first occurs.

To the original poster, I would change the distributor cap and rotor, and maybe even the spark plug wires, especially if you don't know how long they have been on there. It doesn't cost much, and I like to know how long they've been there. It may not make much difference, but it makes me feel better. Of course, that's just me. As for the rotor, it should pull straight off (grasp and pull upwards).

I have the same issue with starting at times, and this is the only thing I have not tried in the past 8 years to fix it (but I will someday):

MrWiretender's Superior state of tune.