What's the intent/gain here? That PCs that pass their Awareness checks go first, then adversaries attack before those that fail the awareness roll?
Ordering rolls highest to lowest - TOR is mechanically normally more interested in pass/fail and 6s rolled, rather than exact totals on rolls. This is designed to speed up play and also makes 6s exciting.
I'll confess to not missing initiative rolls in this system but keen to help!
Making Awareness dictate initiative would make Strength even more important for combat, and indeed make Awareness an important skill for everyone in the company rather than just the lookout. I'm not sure this is desirable - it is nice when different PCs are good at different things, rather than everyone feeling they need to invest in X to do well at combat.
Yes, something like that (but now that you mention it I noticed that enemies don't have Awareness and thus can't make this roll, so my idea is kind of ... not thought through to the end
).
But since I haven't played a single session and just read the rules, I appreciate your experience and recommendation when you say that you don't miss it at all.
And you're right: it would unbalance the game again when making Awareness one of the most important skills (not combat skills) in combat.