To begin, the German embassy checked box 9 on your refusal form. This is...
Your intention to leave the territory of the Member States before the
expiry of the visa applied for could not be ascertained.
...which is a delicate way of saying that your risk of overstay is too great to qualify and in turn this means that you failed a risk assessment. You did not include itinerary and insurance evidence, but they did not address these. Instead the notice indicates that there was a significant document shortfall. For whatever reason, it's inferred that they did not find your evidence reliable.
- The documents you submit do not allow a sufficient economic roots to
be established in Nigeria.
- The documents submitted by you are not suitable for the purpose of
adequately supporting the specified employment / study period.
They also point out (somewhat redundantly) that your ties to Nigeria are not strong enough to overcome the level of risk.
What this amalgamates to is that your refusal is not a case of missing a document or other mechanical issue like insurance, but an overall lack of quality and reliability in what you submitted. This would inform your decision about the effectiveness of making a remonstration because that path is effective for simple problems and yours appears to be an 'across the board' type of shortfall.
A remonstration ('aufzeigen', protest, complaint) is essentially a summary and shortened appeal process. You send them a letter explaining how they should have assessed your level of risk and they will reconsider. At the bottom of your refusal you will find the instructions for making a remonstration, you found a reference on the net instead.
You have asked...
Should I try the Remonstration process via email attaching all the
documents mentioned and including an apology letter for my negligence,
or i should just re-apply?
The answer to this is indeterminate, nothing prevents you from taking either path. If they made a clear error, for example they got your application confused with another applicant, then a remonstration can be especially effective. A remonstration can also be effective if a single piece of reliable evidence can clear everything up. The simpler the case, the more effective a remonstration becomes.
If your remonstration includes a lot of new evidence and written arguments, it becomes tantamount to a fresh application. In that case it becomes more expedient to make a fresh application because the turn-around time is much faster.
The hidden risk in a remonstration is that the clock is always ticking and this means that by the time they get around to processing it, your original premise (along with itinerary and so on) may be out-of-date. If that happens, they will refuse anyway. For example, the surgery has already started, or the reservations are stale, or any other thing could throw your premise out of whack and make your application unreliable. Specifically, in your case, you would be making a bet that they could process the remonstration in your favour by 20 November. Since you already know that they can turn a fresh application around in less than a week, it makes a lot of sense to give more weight to that option.
See also Schengen tourist visa refused. Can I appeal for reconsideration?
See also Schengen Visa Refusal: Does it make sense to appeal?
See also Schengen Visa Refusal: Justification for the purpose and conditions of the intended stay was not reliable
