Outlook not working when one Exchange serve is down

and the other node shows its handling the sessions?
Is the FQDN that Outlook clients the same as OWA? In other words, since OWA is working, Outlook should be using that same pool I would assume.

If so, then you should take a fiddler trace on one of the Outlook clients when that node is down and see what its trying to connect to.
Another test is attempting to bypass the F5 and set the hosts file on an Outlook client of the load balanced FQDN to the other servers IP rather then the F5, then disable the one that was rebuilt in the F5 and see if the client can connect.

Does it fail for all Outlook clients when that one server is down?
The network team should be able to tell if the clients are using the remaining node once the other goes down as the sessions should increase.

Remove the account:

Control Panel > Mail > Email Accounts > View/Change > Remove

Then add it back in.

Have you tried rebooting the LB after rebuilding the server?
You should follow Andy's suggestion that bypassing the LB and connecting the other Exchange server directly by editing the host file on client's pc, if it's connecting, then you can focus on LB or network issues.
Do you have mobile devices using Activesync? Does it behave correctly like OWA?

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Hello, Unfortuantely, we haven't come around a new test.
I noticed the following, when checking the connection status of my Outlook.
I see 8 connections (I have several shared mailboxes). Some of them are connected with the LB name, some are connected directly to the server.

I ran the auto config check multiple times and I get different results.
Sometimes, all the urls' are set to LB name.
But regularly, I see the server's name.