Security architecture is essential in this scenario. As Rodrigo said, you should prioritize. Especially when there's a budget involved. Identify ingress/egress points, identify critical infrastructure that you NEED to monitor without question and try to deploy sensors around that.
You also need to keep data retention in mind. In a perfect world, we'd log full packet capture all day every day but in reality, most networks generate way too much traffic to keep full packet capture for a decent amount of time. Consider solutions such as NetFlow v5 which is much easier on data storage and still provides the data you need to correlate logs and investigate incidents.
Also, the killer, identify where you have NAT to ensure you don't lose half of the logs to NAT. For example, if you have a sensor on either side of a load balancer that is not configured with X-Forwarded-For or Enhanced Logging, source IP addresses will be a lost and investigations will become painful.
When it comes to access to logs, I personally believe that network security analysts in particular should have access to any/all logs available to help correlate logs from various sources and speed up investigations. Not necessarily admin access but, read-only access would suffice. So you'd have logs from your sensors, firewalls, syslogs from routers/load balancers etc. If you only have access to logs originating from sensors, once again, life can get tough.
Final point is to identify all log sources, choose a suitable SIEM and centralize/forward all logs to the SIEM. Jumping to different devices to search a specific IP address between X and Y date is tedious and horribly time consuming when there's a lot of powerful SIEM solutions out there.
When it comes down to it, you mentioned that you didn't want to create a "fire-hose of discarded logs", I personally feel the above is the best way to go about it. Redesigning the network infrastructure from a security perspective will take a lot of work but in the end, you'll be working with what you need and hopefully, noise will be minimal.
Ran over time a little, apologies.