Every jungler, support, and roamer eventually gets good at holding a handful of numbers in their head at once: when Roshan respawns, when the enemy's Flash comes back, how long until the next big camp resets. It feels like a skill you're building. Mostly, it's a tax you've stopped noticing you're paying.
The real cost isn't forgetting
The obvious failure mode is forgetting a timer entirely: missing a Baron window, walking into a Flash you thought was still down. That happens, but it's not the main cost. The main cost is the constant low-grade context switch: some part of your attention is always running a background clock, checking it against the game clock, updating it when something changes. That's attention not going toward positioning, warding, or the fight in front of you.
Multiply that by every timer worth tracking in a given game (Roshan, BKB, Buyback, two or three enemy ultimates) and you're not holding one number, you're running several clocks at once, by hand, while also playing.
What an overlay actually changes
A cooldown overlay doesn't make you a better player. It just gets the clock-running part out of your head and onto the screen, where checking it costs a glance instead of a mental recalculation. Click the icon when you see the ability used; look at the number when you need it. The overlay holds the state so you don't have to.
The part that matters most is that it has to cost less than doing the math yourself, or it's not actually a win. That's the case for click-through mode: the overlay never intercepts a click you meant for the game, so checking a timer mid-fight doesn't cost you a missed input. It's also the case for a hotkey-driven callout: pressing one key to paste "Roshan up in 3:00" into chat is faster than typing it, and faster than doing the subtraction in your head first.
Where mental math still wins
To be fair to your own brain: for a single, simple cooldown you check rarely, doing the math yourself is fine; the overhead of setting up a timer can cost more than just remembering one number. The case for an overlay gets stronger as the number of things you're tracking at once goes up, and stronger still in anything you're trying to call out to a team, where "roughly a minute-ish" isn't good enough.
If that's the game you're playing, the Dota 2 pack and the League of Legends pack are both built around exactly this.