Paying attention to the attitude indicator is absolutely not the right way to fly in visual conditions for many reasons. The best way to deal with reduced visibility under VFR is by doing a thorough preflight planning (e.g. stay on the ground when the weather is marginal) and applying proper risk management techniques in the air (e.g. divert, turn back, land etc when the weather deteriorates)

In the U.S. we are required to give at least 3 hours of instrument training for Private Pilot license. There is no such requirement for Sport Pilot or Recreation Pilot License

In my experience teaching hundreds of private, instrument, and commercial pilots 3 hours is not enough to develop any kind of instrument proficiency. For people who take the instrument rating course it usually takes around 20 hours to develop minimal instrument proficiency. Simulator training helps a lot when done with a qualified instructor who understands how to teach instruments in a sim (unfortunately many don't). But even the students who nail the sim portion often struggle in the real aircraft when they are faced with task saturation.