The problem with padded shoes is either A) the padding makes you slightly unbalanced or B) you're carrying so much weight that the padding gets flattened and becomes worthless anyway. A) causes you to lose power when pushing through the floor. B) destroys your shoes.
Lifting barefoot has absolutely none of those problems and there are a lot of people who do their lifting barefoot for this reason. It's also free. Though most gyms have a problem with this for hygiene reasons (and safety though I'm not sure how a light shoe would protect against falling weight).
Chucks and Vans are often uses because they are relatively cheap, have absolutely no padding, have rubber soles, have flat soles, are low to the ground, and gyms don't cry about them. Though if you find shoes that also fit that profile, then they will also work just fine.
There are also weightlifting shoes which have a slightly elevated heel. This allows you to get a little lower in squats and may help compensate for weak ankle flexibility. They also have very hard, flat, and wide soles which makes them ideal for squats and deads.
For deadlift, there are also "deadlift slippers" which are usually over-priced tai-chi shoes. They have absolutely no elevated heels or padding and are only slightly thicker than socks. This gets you as low do the ground as possible without being barefoot.