To see if it's really worthwhile, you'll want to inventory all of your devices, and what network and WiFi speeds they can handle (especially if they're going through the Eero)
You'll also want to create a topology map (on paper works fine) to draw a line from each device to what it's connected to, so you can see where bottlenecks could be (like you might have a 5 port 1gb switch somewhere, that means you'll only get 1gb uplink back to the router). In a large house there probably are a few switches, small houses usually everything runs back to the router.
You can setup a machine plugged into the router ethernet ports, and run iperf3 in server mode, then run iperf3 on each computer and see what kinds of speeds your getting back through your network to the router.
And finally how do you want to use 3gb? All from one machine, or up to 3 computers each getting 1gb? If those computers are on the same switch, the uplink to the router would have to handle 3gb (so a 5gb or 10gb link), and some of the cheaper switches can't actually handle 3 network ports all maxed at 1gb, so you have to research that too.
BTW: I run 10gb over Cat5e but the run is only about 40 feet so it works just fine. I have a few boxes with 10gb NICs in them though. I work with large photogrammetry/nerf/gaussian splat photo sets and video, so being able to move those around faster is important to my work.
Also, drive speed on devices will affect performance. 1gb is 125MB/sec, which is roughly HDD speed. SSDs are 550/520 MB/s (read/write), NVMe Gen 3.0 are 3500/3200 MB/s, Gen4 are 7500/7000 MB/s. 10gb is only 1250MB/s so twice as fast as an SSD but that's it.
Some future-proofing is nice, but too much over-provisioning can be expensive upfront, and won't be utilized for quite some time.
If the house is wired with CAT5e, why are you using Eero Pro 6s mesh?
There are affordable 10gb switches out there https://www.servethehome.com/?s=10gb+switch
To see if it's really worthwhile, you'll want to inventory all of your devices, and what network and WiFi speeds they can handle (especially if they're going through the Eero)
You'll also want to create a topology map (on paper works fine) to draw a line from each device to what it's connected to, so you can see where bottlenecks could be (like you might have a 5 port 1gb switch somewhere, that means you'll only get 1gb uplink back to the router). In a large house there probably are a few switches, small houses usually everything runs back to the router.
You can setup a machine plugged into the router ethernet ports, and run iperf3 in server mode, then run iperf3 on each computer and see what kinds of speeds your getting back through your network to the router.
And finally how do you want to use 3gb? All from one machine, or up to 3 computers each getting 1gb? If those computers are on the same switch, the uplink to the router would have to handle 3gb (so a 5gb or 10gb link), and some of the cheaper switches can't actually handle 3 network ports all maxed at 1gb, so you have to research that too.
BTW: I run 10gb over Cat5e but the run is only about 40 feet so it works just fine. I have a few boxes with 10gb NICs in them though. I work with large photogrammetry/nerf/gaussian splat photo sets and video, so being able to move those around faster is important to my work.
Also, drive speed on devices will affect performance. 1gb is 125MB/sec, which is roughly HDD speed. SSDs are 550/520 MB/s (read/write), NVMe Gen 3.0 are 3500/3200 MB/s, Gen4 are 7500/7000 MB/s. 10gb is only 1250MB/s so twice as fast as an SSD but that's it.
Some future-proofing is nice, but too much over-provisioning can be expensive upfront, and won't be utilized for quite some time.