Tannus armor aren't tires, they are armor. Tannus has solid "tubeless" tires as well, but those have the problem that you can't choose the tire based on the tread pattern and other features (so studded tires aren't possible), and the rolling resistance is probably even larger than with tannus armor.
I use Tannus armor on my winter wheelset on an e-road-bike. The logic is that when winter arrives, sharp gravel is distributed everywhere to prevent pedestrians from slipping on ice. This, combined with an erroneous idea that cyclists should share roads with pedestrians and not with cars, has led to an environment where good low rolling resistance tires puncture in about 100 km of riding.
My winter wheelset has studded tires plus Tannus armor. The idea is that I don't want to have too many wheelsets, so the winter wheelset is a wheelset that can handle any environment, whereas the summer wheelset is for roads free of snow, ice and sharp gravel. I could of course have four wheelsets: one with studded tires with armor, one with normal tires + armor, one with studded tires only, one with normal tires only, but that would be two wheelsets too many.
The ride accuracy is quite poor, with Tannus armor the bike doesn't go forward in a straight line but the path wobbles a bit and the bike doesn't respond to handling as accurately. Rolling resistance is huge. Normally, I get ~150 km distance from my 500 kWh e-bike battery, but with winter wheelset I only get ~75 km.
My opinion is that you should use such an armor only in extreme environments. Getting puncture every 100 km due to sharp gravel is extreme. I don't believe getting a puncture every couple of months would mean it's useful to use such an armor. You lose a lot of time due to your slower speed. That's probably 20x more time than what it would take to just fix the punctures whenever they happen.
Your "puncture every couple of months" is apparently a puncture every 1400 km, not extreme enough to justify using armor.
In my case, the Tannus armor has reliably prevented punctures. Not a single puncture.