802.1X vs 802.11 – Key Differences Explained

The term packet can be ambiguous, so the standards often use 'acronym soup' to avoid such ambiguity. In particular, often 'packet' is taken to be the layer above 'frame' (i.e. level 3 of the OSI model).

However, as you have found out, that's not always so. Here the discussion is about physical (layer 1) IEEE 802.11 packets. So the frame is encapsulated within the data field of the packet, not the other way round.

802.1x was specifically designed for end-point devices to authenticate to network switches and was not designed for switch-to-switch connections. Because of this, it is highly unlikely that you will find any switches that can be a 802.1x client, so the answer to your Y question is no.

To answer your X concern (see XY problem)- preventing a malicious entity from monitoring and intercepting traffic can only be reasonably accomplished by maintaining physical security of your network hardware and cabling infrastructure. As @Ron mentioned, it is trivial to configure a device to act as a network tap that can be placed between any one of your switches -provided that physical access can be attained- that can then passively monitor traffic on your network, regardless of whether you have 802.1x configured. MACSEC would be an example of a technical measure that could potentially prevent this type of attack.