main hAP ax2 has an EoIP tunnel. The EoIP tunnel interface is a member of that main LAN bridge.
The EoIP interface itself has 1378 actual MTU. For a reason I don't understand, the main LAN bridge was adopting the same MTU - 1378.
The reason is actually very simple. For the router itself, the IP interface is the bridge, so when the routing engine needs to learn the maximum size of a packet it can afford to send to a destination reachable via the bridge, it asks what the MTU of the bridge interface is. Therefore, the bridge MTU automatically adjusts to the smallest one of the MTUs of all member ports of that bridge - which, in your case, happened to be the EoIP one. Otherwise, the packets that the bridge would accept would not fit to the MTU of some of the member ports and get dropped.
I manually set 1500 actual MTU of the main LAN bridge of the main hAP ax2 PPPoE router and this fixed all my problems immediately.
As explained above, it probably hasn't, you just haven't found that out yet - now connections via the EoIP tunnel will most likely fail because the router will be sending 1500 byte packets and the EoIP tunnel will be dropping them. So rather than setting the bridge MTU to 1500, you should have set the EoIP MTU to 1500 and allow fragmentation of the EoIP transport packets and prohibit inheritance of the Don't Fragment bit. This is likely to work (for the price of doubling the packet rate between the EoIP endpoints), but you may still experience packet loss in the EoIP tunnel if something on the path between them has problems with non-first fragments (too many networks do). If this is your case, the only remedy I know is to use L2TP with MLPPP enabled to replace fragmentation on IP level by "splitting" on MLPPP level, which means that oversize payload packets are delivered using two normal-sized transport ones rather than a single fragmented one. So the double packet rate remains but no fragments traverse the internet.
Why these 2 bank websites were not loading remains unknown to me. Why the access to that WINS/Samba shares were not working
As for the banks, there may be a misconfiguration that prevents the PMTUD process from making its job, or someone on the path may do some manipulations incorrectly (i.e. force a MSS value that corresponds to a smaller MTU than 1500 but larger than the one corresponding to 1378 without checking the actual one first), or the banks may intentionally drop connections with unusual MSS, even if adjusted using PMTUD. Regarding WINS/SMB I won't even speculate.