Losing in thought and autopilot kicked in

I think this is fairly common to those people who are observing where they are or what they are thinking. The presence gets better over time.

So, the idea is to not get stuck in the moment when you got lost. Don't judge. Keep going to this moment or the NOW :)

You're still in the present moment because you can't not be. The daydream happens in the present moment. The issue is one of habitual retreating from the world out of sheer boredom. Just develop the habit of looking from the perspective of pure awareness without getting bogged down by the mind. It's really simple but you have to stick with it over many years. The boredom shifts because once you start REALLY looking then you realise that the real boredom is when you retreat into your mind rather than merely sticking with what you've got. Eventually you can retreat into the mind but the awareness stays open such that you can decide whether you want to or not.

Your body and mind recognizes that the path you take to work is easy, therefore you feel relaxed mentally and physically. Try taking different paths that you have never taken before. This process will make your mind more cautious of your surroundings and paths because everything is new.

I think you're supposed to stay aware instead of daydreaming when you're driving.

Try moving your eyes repeatedly to look at different things: ahead, behind, the traffic lights, the other cars, pedestrians on this or that side who might be about to step into the road, etc.

Also breath enough to stay alert.

And think about what someone might do unexpectedly, don't assume they've seen you until they prove it.

I think it's OK to react to changing road and bike conditions without too much conscious thought, but in order to react (even to react automatically) you need to retain awareness of what's happening around you, you need to have seen it, you need to have been paying attention.

When there's an accident it's usually someone else's fault (e.g. a car turning into your right of way because they weren't paying attention and stupidly didn't notice you), but even though it's their fault there's often something you could/should have done to avoid it. For example a common type of motorbike accident is where you come up to a green light, ride through the intersection, and a car coming in the other direction turns across your path (so it's the car's fault), but the motorcycle doesn't even brake before hitting the car (so apparently the rider wasn't paying much attention either).

Also it's not just about you. When I'm on a bicycle I'm fairly harmless and I'm looking out for my own safety; when I'm driving a car I'm fairly well protected but conscious that unless I'm always careful I could really hurt someone else, and that it's my responsibility to never drive where I'm not looking.

Is it really a mistake to be lost in thought?

When Satti is absent you create fabrications hence future misery. Therefore, it is a "mistake" to be lost in thought.

Is my way of countering have bad effect?

You are doing Kayanupassana hence this is one of the solution is being mindful, i.e., mindfulness of the body. In addition be mindful of the bodily sensation when moving like the fabrics touching your body, the wind sweeping over your skin, etc. Also look at muscular pains and pain and pleasure in changing postures.

Ultimately what you have to work with is to realise 4 Noble Truths and Dependent Origination this you can realise through any experience you perceive as good, bad and neutral and this leads to sensation which are pleasant, unpleasant and neutral and even the pleasant and neutral are unsatisfactory due to 3 Characteristics while unpleasant is unsatisfactory on its own and neutral opens us for further existence and it can still give unpleasantness in the future.