The sysadmin cycle of life is to get hired then spend several years fixing, consolidating and automating until someone decides they don't need to keep paying a sysadmin "for nothing".
I'm not a sysadmin but I ended up looking after the office network and you are 100% right about this. Not only that, by the time you know someone has a problem, they're already annoyed, at you, for no good reason. More often than not, it's their own fault.
The other day someone was complaining that the internet was slow (it was), turned out they were syncing dropbox, backblaze and icloud, saturating the network. I understand now why sysadmins lock down everything where they can :)
Thats kinda true actually :-) I am actually a coder by heart and moved now into a sys admin job. I kinda like it because once everything is nicely automated I did my work :-)
Do you know when your sysadmins are doing the job right? When you don't notice.
The only possible way for a sysadmin to get noticed is to screw up.