“In this case your orthodontist can advise you to use another system: Damon or Lingual (Incognito)”
Or SmartClips, or In-Ovation, or Clarity, or Mystique, or STb (linguals), or Inspire ICE, or Carriere, or Time 2, or Opal, or TruGold, or InVu, or NuEdge, or … the comment implies only a couple of alternatives to Invisalign, and yet in terms of bracket brands they are extremelt numerous. More helpful though would be to suggest that if Invisalign is not an option, then instead the prospective patient can consider other _types_ or orthodontic appliance, such as lingual brackets, regular metal braces, gold braces, or ceramics braces. This type of decision is a meaningful one, whereas there’s not really much point to me, as a patient, trying to choose between (say) SmartClips, Carriere, or In-Ovation any more than it would make sense for me to choose my handyman based on whether he uses a Bosch or a Makita powerdrill. The average patient does not have the training or expertise to know the subtle ins and outs of one bracket brand vs. another, and at the end of the day it is the skill of the orthodontist that is of paramount importance any way.

“the need for extractions (or not). Some braces types (Damon comes to mind) might not require extractions, others might do”
The marketing machine at Ormco would have us believe that only through the use of Damon braces can extractions be avoided in certain cases. But the cold, hard truth is that the bracket brand does not determine what results are and are not possible, or what treatment approach is achievable. There is nothing that can be achieved using Damon braces that a skilled orthodontist could not achieve with any of the competing bracket brands. Remember too that results aren’t just how the teeth look and how the bite ends up when active treatment is done, but also include how stable those results will be in the longer term.

“what your orthodontist can provide. Not can provide you with Incognito, Invisalign or Damon. Most of them have just “regular” metal braces (with ligatures), while others are licensed to use all systems.”
Whilst it is true that comparitively few orthodontists offer lingual braces and/or Invisalign, the majority do generally offer some sort of aesthetic bracket option (ceramic, or plastic, or even systems like In-Ovation C), and a great many offer self-ligating bracket systems (In-Ovation, SmartClips, Carriere, Damon, etc.). Again, Ormco’s marketeers would have us all believe theirs (Damon) is a unique system, but there are many competing self-ligating systems on the market and in common use.

“price. Some orthodotic treatments are more expensive while others cost less. Not to mention some are not covered by your health insurance.”
Prices vary country to country, and city to city within a country. What is covered under health insurance, or by your Government is massively variable. However, many dental plans in the USA do not cover orthodontia at all, or cover it only for children. Most who do cover it provide only very limited coverage, with a maximum payout that is generally significantly less than the total fees.

“treatment time: some systems work faster, some slower”
Yes and no. Computer-assisted systems (Insignia, SureSmile) can generally be expected to trim a significant amount of treatment time. But claims from certain manufacturers of self-ligating brackets boasting massive time savings are commonly held to be greatly exaggerated. Again, bracket brand is unimportant.

“allergies. Some people might develop allergic reactions to Nickel.”
This is true, although it turns out that even if a patient is known to have contact dermatitis with nickel, they will often be able to get through orthodontic treatment without issues. Nickel-free brackets (several brands!) are available, and can be used to minimise risks, but there are still the archwires to consider; however, in practice, these generally do not present a problem.

More common is an irritation from latex if elastics need to be used (or, for that matter, resulting from the gloves that are commonly used) However, nitrile gloves are available, and latex-free orthodontic elastics can be ordered, if these are necessary for the patient.

So, in terms of allergies/contact dermatitis, this is certainly something to discuss with your orthodontist at consultation, if these conditions apply to you, but any good orthodontist should be able to work around such issues without any problems.

My advice really is to focus most on choosing an orthodontist you trust, with whom you can communicate well, and whose treatment plan makes sense and is acceptible to you.