A C/L style tournament would be far better than what we have and would feature a group stage which seems to be popular. And not knowing who you might face in the coming rounds - is desirable - especially if you're likely to face a less gifted side by finishing first in your group, thus largely eliminating the possibility that by losing/tieing one (or more, I guess) group stage matches so as to finish 2nd in your group - you'd be better placed to advance in later stages of the tournament, by facing weaker opposition.
But - a group stage isn't optimal if you simply want to provide the best guarantee that those teams who seem to be best going into the tournament don't face each other until as late as possible (e.g., college basketball-style tournament format in the USA).
Other tournament formats are of course possible, catering to various goals - the current format though is, IMO, self-evidently deficient.
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As to your concern about when the knockout stage draw would occur - well - as soon as possible. Problems still exist of course - what if you just recently won your group and now all of a sudden due to the draw you're forced to play against a side with 6 days rest, tomorrow, your side having won in overtime with only 1 day's rest... etc. Well, if you want a group stage, it's going to cause ripple effects - but none worse than the extant format if performed intelligently.
Remedies to the hypothetical posed above of course exist - for example, if a side has only recently played, their next matchup cannot occur until only after a minimum of x-days rest (choose x as you will).