It’s not difficult for me to believe that Alistair, a Templar Warden, or a Templar Hawke were quietly taking small doses of lyrium each day. After all, lyrium is not rare, you’re likely to have a dozen lyrium potions in your inventory at any given time, and it’s not indicated how much lyrium is needed to keep a templar’s powers “active”. So why not go on the assumption that Hawke, Alistair and the Warden just take a little sip of a lyrium potion each morning as part of their daily prayers? Works for me. Then maybe Alistair is just talking out of his ass about not needing lyrium because maybe he’s ashamed of his addiction, or just because Alistair sometimes talks out of his ass.
Regarding the Seekers of Truth, my assumption is that they are actually low-intensity mages – that is, born with definite magical talent that means they can use Templar powers without lyrium supplements, like mages would. In terms of there not being a price attached to this power, we don’t know this at all. They could all be using a simple form of blood magic, or they could all be wrestling with demons themselves. Their use of Templar powers without lyrium consumption could take a terrible toll on their bodies, riddling them with tumours by age 50. We just don’t know the full details of what Seekers of Truth go through.
In terms of why Templar powers work at all, I would say that lyrium, being a magical substance, allows non-magical persons to basically “ingest magic” and use magical powers without interacting with the Fade at all. This is why Dwarven warriors can become Templars. The lyrium is volatile, and the body reacts poorly to it, but through a templar’s mental discipline and training, it can be tolerated and even used as fuel to interact with surrounding magical energies. It has no real interaction with the mind of the Templar, which is why Templar powers are not especially powerful or creative compared to mage spells.