I think you are making the usual mistake of thinking the slippy map tiles are the map; the map is the vector data.
Your first problem is defining what is part of a clean map, and what is not. Although I’ve never used a rendering tool chain, I imagine, once you’ve decided what’s in and out it won’t be difficult to strip all the out things from the tool chain configuration that produces all the in things.
Although I think the funding ran out, there used to to be a site that overlaid a map containing no text, with specific language texts. By a clean map, do you just mean one with everything on the standard layer except for the text.
Note, if people place objects relative to features on an OSM map, rather than based on independent measurements of their location, any publication will, I believe, need to be made available under a licence similar to that for OSM itself.
I want maps to show the basics such as roads, road names, buildings, etc
Some maps look cluttered, especially in city centres there can be big clusters of points such as parking, telephone boxes and traffic lights that are not relevant to the information I want to display.
I’m guessing this type of information would be layered over a simple road map?
That depends on exactly how the style definitions are created. I imagine there is some layering concept in most tool chains, to give priorities, but I doubt it is as straightforward as you suggest. The map itself has no such layering. Any layering is imposed by the rendering tool chain.
As I said at the start, you need to decide what features you want included and what you don’t. A very simple filter can then probably take the unwanted ones out of the rendering data stream .
No rendering has all the features that are actually in the map. All start by filtering the data.
It also depends on what kind of map you’d like to have :
if you want printed map/static image, then use QGIS for example
if you want a web map (slippy map) more or less with what you see on the default OSM.org map, then learn to remove unneeded parts from the code of OpenStreetMap Carto style and serving tiles, probably with additional services (as mentioned above)
There are also different other web map styles, which basically have much less elements than osm-carto, byt might look more like you want it. Then read the instructions attached to them.
They have a number of “background” maps, not all of them show the same amount of data. If you take a rather “empty” one, you can easily overlay it with your own POIs. Examples are OSM Positron and OSM Light. You can find examples of those styles on https://wiki.openstreetmap.org/wiki/Tile_servers