Just a couple of quick thoughts.
When you serve a fault on a first serve, is it normally fractionally long past the service line? If so, it may well be that, because you step forward, you move in too much under the tossed ball (i.e. it's not as far in front of you at contact as it should), so at contact the initial trajectory is fractionally more horizontal than it should.
The other thought is about your toss - and I think it's probably related to the first point above. For me, it looks a fraction too low, when you hit the ball your arm is still bent at the elbow. Seems to me that, because you do end up fractionally too far in and underneath the toss, you actually lower the toss and keep your arm bent to be able to then hit the ball still flat but from (what feels like) right above or even fractionally behind your head.
Tossing the ball higher and further in wouldn't cure this - you'd simply step further forward and over the service line.
I think the cure for this stepping over the line might be if you placed your right foot a bit further behind you (i.e. you turned your back more to the net). This would make your serve motion a bit less linear (forward) and a bit more around, so you catch up with the ball at contact point because your shoulders are rotating, you right shoulder is coming around and upwards and past the right shoulder, and this gives you some momentum also to jump into the court on your serve and follow forward if you want to S&V. Also, as you're rotating more into the toss, you can progressively get the toss a little bit higher and allow your arm to extend higher and straighter - but most importantly, also reaching further into the court - at contact. And as you now hit the ball more naturally slightly higher and in front of you, you no longer need to step forward with your left foot to reach it.
Coming from Bristol, you'll be familiar with cricket - fast bowlers keep their left shoulder forward, but when they get to the crease their left arm comes down to help the right shoulder come through, slinging over and around quickly. In tennis the shoulders rotate on a flatter plane than in cricket, but you still need the right shoulder to rotate from behind and get in front of the left shoulder, to add power from the trunk rotation into the serve. Otherwise, all the power in your serve only comes form the shoulder and the arm.