Air cleaner or not?

I have my turning shop set up in a double garage. I have installed to exhaust fans at each year end of the shop which vent outdoors. One of these turns on when the light is on, i.e whenever I am in the garage. The other is on a timer and runs 8 hours each day.
I am hoping that these draw out all the fine air borne dust.

Having said that, is the still a need or benefit for an air cleaner, something like this?

W326 - AP-12 Two Stage Air Filtration Unit | Hare & Forbes Machineryhouse

Thanks in advance for your wise thoughts

Installing exhaust fans at opposite ends off a shed and having them both exhaust outwards would mean they could be fighting each other for air and dust.
It's usually better to have a large opening like a door or window at one end of the shed and both fans at the other so there is cross draft of fresh air across the work area.

To provide more specific advice it would be helpful to know
1)The size of your shed,
2) the air flow capacity of the the exhaust fans, and
3) exactly how the exhaust fans are set up.
Some photos showing how the exhaust fans are set up would be useful

It also depends on the type of wood dust generation activities.
In general for something like wood turning or drum sanding it's better to extract as much dust as possible at source using a dust extractor rather than letting the fine dust escape into they room and then use a room air filter to try and recapture the dust.

If it was me I be putting any spare $$$ into a dust exorcist upgrading a dust extractor before a room air filter.

As Bob says, the room filters are not the best way to spend your money, but I can confirm they definitely do help. My old work installed several of them (initially 4 of the ones you show and later adding 2 of the giant size ones as well) and the air in the factory was noticeably cleaner; I had no way to properly measure by how much, but it was evident both visually and by smell.