Lol, this is not a “small” one - but just some ideas, I’m sure you will stumble on the cause if you troubleshoot these, at least.
Best I can say is to avoid load balancing, in favor of hash based or failover, until you have the expertise to do it properly - and troubleshoot problems properly. How does your router determine where to route traffic? There are many strategies, and all of them cause different problems… eg. round robin, failover, hash based - it’s just the mechanism by which the router decides which ISP to route which connections to.
Does the same problem still happen if you unplug 2 of the upstreams? Can you identify the broken upstream ISP this way?
Are you using the same public IP address on all upstream hosts? If not, you have to ensure that you have a source routing (or SNAT) rule on your gateway router - so that the traffic will return through the same interface that it was sent from. Many services (such as Netflix) might have security measures to block traffic if it looks like it was “hijacked” - ie. a connection is running from one IP address and then the same connection suddenly continues from another. It sounds like you have an SNAT setup with different outgoing IP’s, but with a local host-based hash - so that the FireTV box is getting its connection to Netflix through another ISP than the other devices. (Even if you load a site like http://checkip.dyndns.org from each device, to check - it might hash those differently from Netflix, for example.)
Also, your main responsibility is establishing end-to-end connectivity - so a good first step is to ensure that the “not found” page is coming from the remote site, and not from one of your local devices - eg. Netflix might have policies in place to block certain devices - the way to deal with that might require more knowledge about their policies and how to work around them with - by routing through another ISP, or using a VPNs and so forth - but as a “network service provider” you’re not obliged to help with that - unless it is actually your network setup that is interfering with the connection.
So you can start by adding firewall logging rules on your exit point - and to confirm that the traffic is actually leaving your network, and returning from the remote network, and also that it’s doing so on the right connections.
Some networks might have explicit or transparent proxies, and they could be misconfigured, or set to only allow traffic to- or from- certain devices. The upstream ISP might have the same - in which case you need to raise the issue with them.
Then, there might also be MTU problems - some routers or devices might be outdated and dropping certain connections because of the packets being too big, and some responses might not be coming through. So a last resort would be to check the path MTU, and lowering the MTU on some of the ports if you see that certain connections are getting broken.