Very true, most methods are "just OK." Any value needs to be interpretted considering all factors. Ambient temp vs fluid temp, distance from measurement to item (seal,cooler)if heat transfer from pipe could be a factor.
I will argue that a differential temp reading is more valuable in this context than knowing exact fluid temp from a single temp gange. The most important information is component temp and relative temperatures; temp rise across seal, temp drop across cooler.
These will indicate any problem with components, or any condition that is different then "normal operation." Really, you are looking for something to be "hotter than usual." With one temp reading, the cause could be high ambient temp, high cooling water temp, high process temp, rub in the seal, or fouling of cooler.
A couple stick on liquid crystal thermometers will tell you who/when/where and reduce the what/how to something that can be answered by anyone with a simple troubleshooting checklist.
You could even stick them on an employee's forehead to see if they need to take a sick day, or in their desk chairs to see if they've spent too much time chatting at the water cooler!