How do you think this impacts the player's ability to strategize?
Not at all, I'd guess, except they have to convince the DM that they can take an extended rest here and now instead of running out the clock. Giving the DM that extra bit of responsiblity in making a judgement call might be a good thing for the game.
Remember, the goal isn't to shaft the players.
The goal is to prevent/avoid the one encounter a day fights that the players are otherwise entitled to. I completely understand that players won't want to give up an advantage if they don't have to, despite how the game becomes objectively worse (because one encounter a day fights are robbed of excitement and strategizing of resources).
If the rules already in the first place support the notion of multi-day strings of encounters then the burden of voluntarily abstaining from an advantage doesn't fall on the players.
Now, with that out of the way, to your question...
Yes, if the DM (ab)uses this new rule to always keep players in the dark as to when and where they might take an extended rest, everything becomes guesswork and something (important) is lost from the game.
So I believe the DM should be clear on what conditions need to be fulfilled in order for the PCs to take their extended rest; as much as possible (without completely breaking world immersion).
As for the desert trek, the DM can tell the adventurers how many days they think they have left until they reach the next oasis (or whatever). Of course you can have seven encounters in one day, or you could have gotten lost in the desert (adding traveling time), but ideally, there should at least be hints that point towards these facts for purposes of resource conservation ("you're probably lost, so hang on to your dailies and surges").
As for the Mummy's dungeon, I think it works best if the DM tells the players outright how many extended rests he's allocated ("you have time for one meal, so that's one extended rest").
Different adventures require different amounts of extended rests, and this way makes for a tighter integration between rules and adventure. 4E is very gamist anyway, so allowing two rests in the Catacombs of Chill while you get five in the Haunted House of Horrors (despite how there might be no "reality"-related reason for the difference) is not a problem in my view.
Allowing the players to "convince" the DM to give out extra extended rests might work too, of course - it's not a board game, and leaving things up to the DM will by definition be for the best.
However, as long as the game does not impose any kind of penalty for taking an extended rest (except the incredibly vague notion of "some adventures run by the clock and resting might have consequences"), I think this should be used sparingly.
If at all. Games-wise, what's appropriate here is to use in-game tools to procure extra Extended Rests. Don't remember its name off-hand, but there was one ritual that gave you the benefits of on additional extended rest.
Just begging the DM for extra rests should have consequences, and ideally these should be codified by the adventure. Perhaps one extra rest adds additional monsters. Perhaps you need to concede defeat for the second (that is, by taking that second extended rest, the Dark Lord eats the princess, and you will have to move on to the next adventure).
What I mean by all of this, I guess, is that begging the DM for more extended rests is hardly an interesting strategy. Strategy would instead be what you have today ("should we explore these rooms or go straight for the Mummy?") only expanded into all kinds of adventures.
Did that make sense?