Be aware: you want to reproject your layer or change the CRS of the layer? That are two fundamentally different things and the linked post presents options for both: that could lead to confusion.
Normally, you should not change CRS, but reproject (click the link to see why).
To reproject multiple layers at once, run Menu Processing / Toolbox / Reproject Layer and check the box next to Run as Batch Process… (bottom left). Fill in the dialog window that opens (see screenshot).
If you really want to batch-assign a new layer CRS, use Menu Processing / Toolbox / Assign projection and again run it as batch process.
Projections are not part of the style, that's why you can't use the approach with Copy style.

Edit
By the way: don't confuse layer- and project-CRS, that are complete different things! You write:
...30 layers. Each of them has project CRS EPSG:4326, which requires
change to EPSG:3857.
If you want to change the project CRS, simply do that at the bootom right of your QGIS window, no need to change anything for the layers. Make yourself comfortable with the difference of layer- vs. project CRS, see here for details.
So basically, you should differentiate between:
- Reproject a layer
- Assign (change) a layer CRS
- Change project CRS
Menu Processing / Toolbox / Assign projectionin batch mode, see updated answer. – Babel Jun 30 '21 at 14:46EPSG:4326, changed toEPSG:3857, will show a point with geographic coordinates 10° E, 45° N (=northern Italy) somewhere in the Atlantic south of Ghana: coordinates are interpreted as 10 meters east of the prime meridian and 45 meters north of the Equator. That's probably not what you want. That's why you should reproject. – Babel Jun 30 '21 at 16:34