Local network blocks outgoing packets from devices occasionaly

You have done your troubleshooting well. By observing the problem without any routers in the mix, you've shown that your trouble is in the modem or, more likely, the ISP network behind it.

One more thing to rule out: completely disconnect your routers from your modem, switch them off, and repeat your test directly connected, with Ethernet not WiFi, to the modem.

If you still have the problem you have proven the problem is with your machine, the modem, or the ISP network.

If the problem goes away in that situation it's possible that some other machine somewhere on your network is occasionally blasting data at the modem.

Four routers sounds like an overly complex setup. The typical modem connects to only one router. Routers actually route packets from an upstream network (your ISP) to a downstream network (your building's LAN). If there's confusion about which router is supposed to do that, frankly it's amazing that anything works. If three of your four are configured as WiFi access points that's OK.

The typical modem doesn't offer DHCP service to its downstream LAN network, because it doesn't know about a network behind it, only one device, your router.

Some ISPs provide combination modem / router boxes. Maybe that's what you have. You may want to review your network configuration.

Try tracert command in the command line to see if your packets are travelling through all these routers. Maybe it will solve the problem or at least displays if the problem is caused in the routing. Or try to see if the problem is caused in a wrong default gateway to the device.