Your friend tells you they're a paedophile

[Edwardo Piet]

If I truly trusted this person [then] my kids would have nothing to fear from them, nor others supposedly.

Non-sequitur. Doesn't matter how much you trust someone it doesn't indicate that your kids have nothing to fear from them. It's just subjective trust and when in the face of expressed urges that could potentially lead to harmful behavior no amount of subjective trust is worth the risk of you being wrong about the person.

[...]encourage them to admit they have a problem and seek the appropriate help.

If possible and realistically achievable, yes. But I don't think it is.

In principle it's a good idea: if we could just encourage them to admit it and seek help as a society then that would be the morally good thing to do because they'd get treated before we had any evidence of any crimes committed: but in practice I don't think it's a realistic society that we could live in.