Not as easy as all that.
I disagree with @Catprog and @o.m. - especially given the time-frame.
A lot of material is going to get taken out to sea via weathering processes since it's now exposed on surface (and if metal is correlated with humans, most of it is near the sea: In America it's 39% of the population). Most of the ocean stuff will have washed away in hurricanes, global warming, etc, etc. eg: NYC without pumping running ends up underwater / subsiding. Anything that gets dissolved / taken to the sea will precipitate out of solution... onto the ocean floor. You'll then have to wait for it to go through the subduction zone and then recycled up into ores. That'll take a minute.
For an example, where is all of the gold, bronze, etc that was mined in ancient times? King Midas was a thing, since he had a mountain of gold ore. It got worn away, since the West used it in coinage.
Or, put it this way, how many garbage dumps from pre-Roman times have metal in them? Or even Roman midden-heaps?
Over thousands of years (or, heck, hundreds) most metals will be scavenged by primitives and used for knives, etc. Worn out and discarded. We don't have any bulk metal from civilizations that're as old as your proposed timeline.
Bronze was never easy (nor cheap), and required some world-wide trade routes to get at the tin necessary. It was too expensive for most of the things that needed to get done. How much tin do you expect to find in a garbage heap? How much tin (or bronze) is in your house right now?
Any of the stuff left isn't going to be that pure, most of it will be oxidized, or perhaps reacted with acid rain, or other by-products. You're going to have to figure out how to run smelting to get at pure materials. You can do it with charcoal, but you'll need forests* (4 to 1) in the right places. You won't be shipping anything by rail, without huge iron deposits to make those rails with - which means any 'garbage mines' will have to be by a river, ocean, or canal, or will have to be located in a forest.
Anything that's corroded, oxidized, or contaminated can be worse than some of the ores we've had. Granted, some of the ores we have had, and used, are exactly that (oxidized metal) - but they're often not the best ones.
All of these would make metals even more expensive than they were historically. Maybe much more. Which means you're not going to be doing a ton of experimenting with them. Nor making experimental devices that blow up out of your (equivalent of gold-reserves). The more expensive something is, the less likely you are to experiment with it.
- How're you going to have trees/forests? If you have a catastrophe, you're going to have to wipe out enough humans that they don't burn down all the trees for warmth when the first winter hits that they don't have industrialized society to run their heating for winter. If they haven't already cut them down to cook their food. And they'll need to not be wiping out the remaining trees when they harvest for next winter, and not wiping out those remaining trees for next winter... until you can get new growth to mature enough to be reproducing (and outpacing new human growth, and survivor's attempts at restarting industrial civilization).