It's a ELS v8 bug.
The default siren (Wail, WL / M1) doesn't stop traffic (that is, traffic does not move to the side of the road when siren is on [if said traffic vehicle is moving at speed] ), if you switch to siren2 (Yelp, YP / M2) or siren3 (A1) & verify that traffic stops, after that the traffic will stop for Wail (WL/M1) siren too.
The traffic does seem to randomly stop stopping for Wail (WL/M1) siren again, & you've to again switch to Yelp or A1 & do the procedure again.
I just wrote what I know about this bug, based on what you said & I what I think your problem looks like.
I don't know what Left Thumb does on the XBox controller, so I might've misunderstood your problem.