I don't think Chyrsler or Lamborghini put enough money in to building an F1 program, development was a little slow compared to other teams, and they tried to put their engine into too many teams (at times).
In 1989, the enigne had all the teething problems a first year is going to have, but it sure was quick when Alliot drove it. But in 1990, the engine was much more reliable, and the Larrousse (or Lola, as it turned out) was a reliable and efficient car design.
But for whatever reason, the engines went to Ligier for '91, and the car was a dog. Ligier didn't care becasue they knew they were getting Renault V10s the next year, so they didn't encourage any development of the Lambo unit. On top of that, there was some emphasis on that Modena/GLAS project (which got called Lamborghini) and was just a money pit.
In '92, the engines went back to Larrousse (but strangely, the chassis was called "Venturi") and stayed until the end of 1993. More power was developed, but reliability wasn't. It would have been interesting if McLaren had taken over development of the Lambo engines! But then again, it might have been another Peugeot.....
Alliot was a steady journey driver at best, but yeah, he was never given anything remotely competitve to drive.