> Real, filtered table beers
Who said anything about "filtered"? We're talking about watered-down beer here (or rather, to put a fine point on it, beered-up water.) In other words, grog, but with [whatever spirit is commonly produced locally] instead of rum.
> The issue with spirits and table beer was it needed heating, and that was costly.
Let's walk through the lifecycle of a small beer here, because I think you're conflating a decentralized economic logistical workflow as if it was all the responsibility of one dude.
A local brewer/distiller makes high-proof beer/spirits. Which does require heating, sure, but just once, when making the batch.
Then pub/tavern-keepers buy those high-proof spirits, by the caskful. They can let them sit around forever until they use them up, despite the casks not being particularly sanitary, because these wholesale spirits' high alcohol content and low water content prevents spoilage indefinitely. So they're good investments, something they're willing to put money up front for. (And these large infusions of revenue — perhaps even on a predictable contractual basis, for something like a tavern — are how the brewer stays in business, despite the expense of burning wood/coal to brew.)
Each day, any given tavern-keeper might fetch a barrel of fresh-ish creek/well/lake water. Might be infectious, might not. Even fresh creek water can have Giardia.
When a tavern keeper wants to serve someone small beer, they take a mug of the creek water, measure out a jigger of the high-proof spirit, and stir it in. The alcohol content kills whatever might be in the water, and if it's something like beer, probably maybe makes it taste better, too.
The creek water will spoil in a tavern-kitchen environment, but that's fine: it's just water. Treat it as grey water at the end of the night — use it for dishes or whatever. Maybe take some for boiling puddings in. Dump the rest on your garden.
This is basically the same logic as fountain drinks. A central provider spends a bunch of ahead-of-time effort to make a bunch of concentrated syrup that's too high-sugar and low-water-content to ever spoil. Then restaurant owners buy this, and combine it just-in-time with water to produce drinks.