So, here's my take.

Anytime the device changes APs, there will be a brief blip when it switches.  You probably wouldn't even notice it unless you're walking around with a smartphone on a Zoom call or something.

Without client steering, if the client thinks the signal is weak, it will start looking for other APs with the same SSID.  ...and usually, that's fine, especially for devices that roam the network like phone or a smartwatch, you probably want that device to maintain connection rather than switch networks frequently.  

With client steering on, you're having the grid monitor signal strength across APs, and even if your device is perfectly fine continuing to broadcast to the same AP at 60% strength, the AP will tell the client to stop and use the one that's at 90%.  Connection interruptions are still very brief, but more frequent.  Even for stationary devices like TVs, if something is temporarily in the way, Eero will ask the TV to switch, and then switch back again when the obstacle is cleared.

What frustrates all of us as users is seeing that the stationary TV set has for some reason received such a low signal strength from the physically closest AP at some point that it connected to an AP that's far away, and didn't come back when the problem cleared.  Client steering works fairly well in these cases to redirect the device back to where the signal will be strongest.  

But fair warning, "strongest" long term is more of an art than a science.

In summary, with client steering on, you're sacrificing connection stability to get higher signal strength.  With client steering off, you're sacrificing some signal strength in favor of signal stability.  

It just depends on what's most important to you.  I leave mine off, and that's just fine for my family.

Oh, and as others have noted in other threads, for IOT devices, Eero sometimes will tell them, "you know... there's a much better AP you could be using," which the IOT device responds with, "that's nice. I'm fine here."  🤷‍♀️

I just posted something similar, I just set up a new 3pack eero 6 pro.  When I go upstairs it's stays connected to the downstairs router and vice versa.  After an hour I still see the same.  I get a lot of lag on YT videos as a result and speedtest show 30Mb/s download.  With TP-Link X55 which I returned thinking the tri-band 6 pro would be better, it handed me off to the nearest access point within 1 minute, and it was consistent.  Can they really call this a 'mesh'?  I am going to return this POS if this is how eero works.

I used my phone for the Tp-Link and eero so no device issue.