Yes,, as Chalo pointed out, slime screws you for patching tubes. But sometimes a big nail will make you have to patch.
If I really need to patch in the field, step one is get the hole high, so as much as possible slime drains to the bottom. Remove as much air as possible, so no slime is trying to get back out through the hole. Clean as much as possible, carry a rag for this. If really slimy on the outside, which is common, wash with dirt, then wipe dry with the rag.
Then use a large patch. The patch may still not stick well, so you need more area glued down. Not a permanent fix, but it should get you home.
On a really critical ride, like I'm going to be 30-40 miles from town in a lethal desert, I definitely carry a tube, and a spare tire! But if a hole in a tire is relatively small, say less than a 3/4 inch cut, then you could carry a piece cut from an old tire. Stick it in the tire over the hole, re inflate to a low ish pressure, and limp home.
For a permanent patch on slimy tubes, you get all the air out you can, drain the slime away from the hole, and wash the area enough to get a permanent patch to stick.
Back to the sweeping. Oh yeah,, that's really funny, ask the city to sweep. Actually the place I have problems the city is not allowed to sweep. It's the frontage road of a state highway. My city actually does a great job of the sweeping, particularly in the rainy season when sand and gravel is washed into the streets. Some years, I'll see the sweeper in my area twice in the same year. But the state highway,, well, once a year would be heaven. Once in three years is more typical When they do come sweep the wide bike lane on the frontage, invariably the guy is driving along with his hopper full, so the sweeper basically just taking the accumulated rocks, dirt, and nails in the gutter, and spreading them evenly over the entire bike lane and part of the car lane. Thanks bro.
Soooo,, The corner by my house that my car, my wifes car, my bikes and my scooter travels daily, I stop about bi weekly, and pick up the nails and screws. Farther down the road, I stop and pick up nails in the intersections when I see them. May as well, they will be there for years, where the cars flip them into my bike lane.
And,, I actually sweep and shovel the first 2 miles of that road annually or so. Nobody else is going to do it, and that first couple miles I travel often. It's the route to a dollar store, veggie store, and the flea market.
I did rig a broom on the bike once, and it did a good job knocking the larger rocks to the gutter. But really did not do a thing to nails lying flat. I still need to try a deflector right in front of the rear wheel for nails. Maybe a magnet on it.