Has anyone got experience or thoughts on the Tannus Armour inserts or airless tires for a road bike? I am getting a flat per month and it’s getting tiring 🤣.

Flats from what? Unless they're large punctures (<1/4"), this is exactly the scenario where tubeless shines.

What's causing the flats? If pinhole punctures, go tubeless. If rim tape, fix your rim tape. If pinches, run higher pressure and/or stop slamming into potholes.

Small sharp glass, sharp pin sized metal, no big hazards. So tubeless is the way to go?

Yup.

Your wheels/rims need to be compatible, though, so hopefully they are already.

In terms of sealant, don't use Stan's. Use Orange Seal or something similar that's suitable at road bike tire pressures.

If you're going to do the work yourself, just be aware that it may be difficult early on - there is a learning curve - but it will get easier.

What tires are you having the flats with? I used to run some tires that would pick up glass, rock shards and other things in the tread and they'd stay there but not so bad that they immediately punctured the tube. Before each ride, I'd inspect the tires and remove all those potential flats. I'd go months to over a year without flatting. Times that I didn't inspect them for a few rides resulted in flatting.

Changing to a different tire eliminated those things sticking into the tread and I no longer even inspect my tires for shards before riding. Vittoria Rubino Pro's of a version no longer made and my current Continental GP 5000's don't pick up shards like other tires I've owned way back when did. And I still go a year or more without flatting. Though when I do get a flat, it means at least two more will happen in quick succession no matter where I ride.

But yes tubeless might be the thing for you. Either way, you have to try something different than what you are doing now.

My daughter was getting all kinds of flats in LA at the time when tubes were hard to get. She put a set of tannus tires on her old steel Schwinn and loves them. If your bike is your only means of transportation and you commute in the city they can be a good thing.
sometimes dependability and durability outweigh performance.

In September I got a new set of Continental Grand Prix 4 Season tires 700 x 32mm. I live in a semi rural area and don’t think there is a street sweeper in the entire county so stuff just stays in the bike lanes.

The sealant point is especially poignant for me; I struggled with tubeless basically just coating my frame in latex every time I hit a goatshead because I was using Stan's.

Orange Seal appears much more resilient, though anyone who is sealing a 1/4" puncture with sealant ... I want to know how.

1/4" has been the break point for me - they'll typically seal but are prone to breaking back open if/when you hit a bump/crack/seam just right.