It seems like the cooldown indicator only shows up when the pickaxe is trying to mine an empty area or if you send the attack/destroy input in quick succession.
I am guessing that the mouse button being held down is treated as one continuous event whereas a key being held down is more like multiple events being fired extremely fast. I could somewhat verify this since rapidly clicking the left mouse button produces the same cooldown indicator as holding down a keyboard button, whereas holding the mouse button down does not.
Personally it didn't feel like this was affecting the rates because the basalt got instamined as soon as it generated. But if your farm design is showing a significant difference, you can try the "Reload textures" trick as described in this answer to make the program think you've kept the mouse pressed. Only difference being you'll use the left mouse button, and for me it worked even without Alt+Tab as long as I let go of the button while it is reloading.
I built a simple, no redstone farm in Minecraft Java 1.19.2 by following this YouTube video. I tested it both in the overworld and the nether (faster lava flow) with an Efficiency V Diamond Pickaxe and a Haste 1 beacon to achieve instant mining. I did spot some more complicated redstone designs, maybe they force block updates to occur faster and that is why you are experiencing a difference.