Chest ECGs done at the hospital are often 12-lead, and those give different "angles" through the heart, which gives cardiologists more clues on exactly which parts of cardiac tissue are causing a problem.
A one (two?) lead ecg as you describe is not going to tell anyone what is going on with ST segments, particularly as you would be only looking at your aVR or aVL(depending on which way you sum the currents).
An electrode on each arm gives you Lead I.
Lead I is not as useful as Lead II, but it's still plenty useful for diagnosing all sorts of cardiac issues (there's a lot more that can go wrong with your heart than just an MI).
To be clear, by "lead" here, I mean a pair of electrodes. A single electrode (like you might put on a watch) is useless.
(I used to be an EEG tech, and for us, the ECG was the primary thing we wanted to get rid of...)
Most clinical EEGs are related to epilepsy or similar, which usually has a particular pattern with spikes in it, but sometimes that pattern is subtle in the EEG. The spiky nature of ECG 'bleeding through' can make it a bit harder to determine the true EEG spikes from the artifact ECG patterns. Usually you will record a single-lead ECG along with the EEG, so you can see where the ECG spikes occurred and account for them in diagnosing the EEG.
This link has some clear spike-and-wave epileptic waveforms in it, but it's not always so clear-cut: https://en.wikipedia.org/wiki/Spike-and-wave
Finally, there's not a lot of overlap between neurological and cardiac patients - there's rarely any need for an EEG tech to do a full ECG (we never did, and we did approx 3200 patients/year), and full ECGs are pretty commonly done in most other places in a hospital anyway.