I think, although Steve might know for sure (?), that the reason was that from about 1975 there were no new NEP/DEP pilots coming into BA on the Trident, so by about 1980 the minimum co-pilot experience was 8-10 years or so, and the CAA, who had vetoed it before, were persuaded by BA, with this amount of experience, to approve the tridextrous route - economics presumably being the driver.
I don't think it was the 4th engine, all this did was give extra thrust for T/O and although its operation was complex, it was just another system. I think the main reason was the very different pitch attitude on approach. IIRC 7-9 degrees in a T1/2 and 3-4 degrees in a T3.
I had done 2 years on the T1/2 when the T3 came in. Initially it attracted more pay, so all the senior guys went on it. After a short while there was parity with the T1/2 so nobody wanted to go on it - it didn't go to the Eastern Med - so juniors were posted. After 7 more years on the T3 I became tridextrous, 1 sim session in the T1 sim, 3 sectors line training with two landings. IIRC you couldn't do 3 consecutive sim checks in the same sim, and there were no requirements for recency flying on the 3 types.