Street names from OSM are reducing when a map is exported, the end result doesn't match the preview. Any advice on finding a solution in QGIS 3.16.9?
The bad news are, that there is no solution. You have to play around with the resolution (dpi) of your output (Properties->Layout Tab->Export settings->Export resolution). The higher the resolution will be, the smaller the labels will be (see https://gis.stackexchange.com/a/305658/87346 for example). If you want to get the same result as in your mapcanvas, you have to use screen resolution: 72dpi.
The good news are, that from QGIS 3.20 upwards you will have a preview of the xyz tiles, if you change the resolution of your output, the xyz tiles will be updated to that resolution and you will see what you get: https://www.qgis.org/en/site/forusers/visualchangelog320/index.html#feature-improvement-to-xyz-layers-previewed-in-map-items
To get larger text sizes from WMTS/TMS you can set the maximum zoom level to a lower level (i.e. 10) or reduce the export resolution in layout manager:
You could also use GDAL-VRT to lock levels and enhance the quality of the output: