I'm one of these people who screens on this subject prior to going into a new relationship. But only to a degree (I get the gray stuff).
For instance, I learn through dating whether a man is still involved with an ex. If he tells me that they're still best friends, I thank him for his honesty, and that's our last date. If, like you, he's kept in touch during Covid or he still has a business relationship, or whatever, it's a case-by-case observation, but I'm not inclined to involve myself there. If he co-parents with an ex, then it's on me to observe over time whether their relationship is something I can live with. I'm inclined to be fine with even the friendliest of contact. It's the adversarial stuff I'd have doubts about, because to me, when couples aren't behaving as a good co-parenting team, it signals that one or both are using their kids as a battleground to continue their toxic attachment to the other.
But once I clarify for myself the degree to which a potential lover is involved with an ex, one text would certainly not derail me, especially given that he's straight out told her that he's with me now.
That's the thing that I would stress to your GF, but if she's still mistrustful, then you're not going to be able to fix this. There are two kinds of jealousy: the kind that's based on suspicious behaviors--but those are plural and raise concerns because the pattern of a person has become un-trust-worthy. The other kind of jealousy is the toxic stuff that someone brings into a new relationship and projects onto the new person. That's battery acid, and it signals that this person isn't relationship material.
There is no proving a negative, and there is no reason to remain with a person who cannot offer you trust as a foundational aspect of a relationship. You'll never have that with someone who projects mistrust from their past onto you. That's stuff they need to be left to work through with a qualified therapist.