You’d be surprised. I thought the same and always cover bars and headset during trainer rides. Took apart my headset this weekend and a lot of salt and gunk was in there.
I have the triangular sweat catcher, and wear gloves indoors just for the sweat. I’m a pretty heavy sweater and find this works well. I find this much more comfortable than having a towel over the bars.
I steal hotel towels for this purpose 
I bought a stack of washcloths for this purpose. Soak them through, reach for another.
Trust me…you do. it doesn’t take much and you’d be surprised how much accumulates over time. Sweat is incredibly corrosive.
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I tried those bike/handlebar sweat covers and while it may look like it works, its when you stand up on the pedals it starts to chafe the inside of your legs. Also depending on your fan placement, sweat tends to drip in other locations like behind you, on the trainer, BB/crank, etc. Those specific handlebar/bike covers are pricey at best and not movable or adjustable as I would prefer.
If you have a front or side table with your tablet/computer and with the fan(s) placed correctly, you should be able to keep cool and most sweat off the bike with a smaller gym-size towel, or better yet, I found a golf towel perfect for this. Yes I said golf! 
The center slit goes over the stem area and when I anticipate on standing on the pedals, I move the back of the towel up a bit on the top tube. The top or front of the towel is draped over the front wheel/riser to prevent sweat from hitting those areas.
Here’s the link to the golf towel which I think works perfectly…
One last note in addition to the said towel above, having a headband plus wrist band(s) helps to prevent sweat going on the handlebar and front areas.
A simple solution, which I use, is to use 2 velcro straps (see the below picture) to temporarily attached a rolled up towel to the top of your handlebars. This keeps the towel from moving, protects your bars, and doesn’t block airflow from your fan. Plus you can use any towel you have lying around, so it saves $$$ as well 
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Don’t get sucked nto marketing gumph, towels and a wooden blocks are more than adequate as sweat catchers and front wheel risers.
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Would an old pair of socks work. Cut the toe off, and thread over the brake levers.
Probably. You can also use hand towels and wash clothes. They are very inexpensive in the US at Walmart.
Zoom in on my handle bars for my setup - one wash cloth each on the hoods and stem (3 total) and a hand towel across the aero elbow pads. I’ve tried one of those triangular contraptions but it rubbed my legs, and was way more expensive than cheap wash cloths.
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Sweat bands and a tea towel!
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