I'm afraid I totally disagree. Saying "Continue Approach" supplies just as much situational awareness to other crews on the frequency as saying "Continue to Breath". Other crews are totally clueless about the position of this particular aircraft other than it's "out there somewhere".
I can see it from both sides. When a pilot checks in on my (tower) frequency and I can't (or won't) clear him to land, "continue approach" emphasises I want him to continue normally. Roger is just a bit vague - but you of course can acknowledge my "continue approach" with "wilco", or just "continue". If there's a reason I want you to "continue", I'll give it, i.e. "Continue approach, one to depart" or similar. If there's an issue, then I'll also state it, along with any action - such as "reduce to minimum safe approach speed", to prepare the pilot for what may end up being a missed approach. This is becoming more regular now we cannot depart anything after a reported birdstrike (or suspected birdstrike) - and it always happens when it's tight and you've given the one at the hold a conditional line up clearance...
Finally on this subject - my bugbear! If we're quiet, you check in at 10 miles, nothing's going on...you will probably get a "continue approach", not a landing clearance. Why? Because at 2 miles, you'll ask "can you confirm we are cleared to land". Thanks for getting confirmation - I'm not against that at all, but invariably it gets us jumping up with binoculars wondering what you've seen on the runway! If I just give you a continue, then a clearance at 4 miles, I don't get the raised heart rate! I often wonder why the flight deck get a placard for the cabin crew to set "cabin ready take off / landing", but there isn't one to say "cleared to land" when landing clearance is received...