I would say that the DMG is trying to oversimplify this so it can be adjudicated without effort or extensive knowledge of sailing on your part as the GM. Primarily because those rules keep the ship out of the way of your PCs' grand adventures.
The official rules are giving you a high-level abstraction from the day-to-day effort of naval travel. Is that abstraction helping or hurting your game? If it is hurting, then ignore those rules and adapt, either via hand-waving an X% multiplier to travel times or via more complex adjudication. (Compare this abstraction to the use of HP to abstract fatigue/damage in fights.)
in a more "real-world" scenario, there would be some mobility at crew levels below those minimums, but with severe limitations and possibly greater risk of errors. For one thing, a ship that's out in the deep sea probably can't just drop anchor and wait out the night; someone has to stand watch. And a reduced crew would have to work harder to maintain the ship, so they would become fatigued and more prone to errors. Plus, all those "make-work" tasks a full crew can perform would begin to slide, since every day would involve some prioritization of work in which something would get dropped as not important enough. So for an emergency voyage, a voyage after a bad storm that kills some crew, or a run after a devastating fight etc...
As a GM I would say the ship can make emergency sail but will be at greater risk in several areas:
For each day of travel, I would do something along the lines of...
- Greater risk of crew injury from fatigue/overwork (including falling overboard)
- Greater risk of "damage" to the ship from improper maintenance
- Reduced speed
- Reduced combat effectiveness up to and including being completely unable to mount a defense from other naval attacks
- Much greater risks of crew harm or ship damage during storms or etc.
- Reduced maneuverability (up to and including inability to travel against the prevailing winds/currents)
- Possibility that certain normal tasks cannot be performed due to loss of specialized crew. (In other words, don't lose your navigator.)
- Any emergency maneuvering would immediately risk the ship and crew, since there aren't enough people to do the job properly
- Any action that a ship could normally do (not just travel times) would require more time to complete: docking, raising or lowering sail, dropping or raising anchor, etc.
I'd also suggest that over time, those negatives begin to get worse. This should encourage a competent sea captain to head to safe harbor with whatever haste they can make. I'd also say that at some point, you will reach a minimum that restricts you to going where the currents and/or winds take you and not much else. I don't know enough about sailing to know what that line is for various ship sizes. The bigger the ship, the more people it will take to not make it a deathtrap.
Also remember that crew members are not interchangeable parts. Googling naval titles indicates there was a great degree of specialization within a typical crew, from untrained hands to highly skilled navigators, carpenters, etc. This site has a pretty good rundown of common ship's duties. So if those critical crew are the ones you've lost, you are in more serious trouble.
Granted, that means not just keeping a head-count, but then tracking who is doing what, so when the crew get killed, you know what knowledge is lost. That may be more bookkeeping than you want to deal with. But if your campaign is intentionally a naval campaign (and the ship isn't just a more complex horse and buggy, to get you where you need to go), then this may be necessary.
Oh, and as for dropping anchor at night, remember that the anchor is raised by crew members via muscle power. On larger ships, the anchor is on a tackle, so there's some multipliers to help the lift. But if you're on a large ship and you're down to 3 or 4 people, you may not be able to raise the anchor at all. An anchor works by dragging into the sea bed. So it can't work if you're in water deeper than your anchor lines. And if the wind changes while the crew sleeps, then your anchor may no longer be set correctly. So yeah, the risks go up as the crew goes down. This link lists a few common anchor mistakes.
On the plus side, if the ship had proper stores (food and water) for the ship's crew to travel X days, then as the head count drops, the voyage has spares now! Yay.
Given the width and breadth of D20 supplements out there, I bet there's a naval source book that can dive deep into this material if that's something you want to explore in more depth. (puns intended)