Armenia

It’s considered a level 3 due to COVID: https://travel.state.gov/content/travel/en/international-travel/International-Travel-Country-Information-Pages/Armenia.html. Your passport must be valid upon departure and no visa is required. I would check Armenia’s COVID restrictions for the latest and enroll in STEP to be informed when things change: https://step.state.gov/step/. Lonely Planet released a guidebook in Jan (7th edition): https://shop.lonelyplanet.com/products/georgia-armenia-azerbaijan?via=Z2lkOi8vbG9uZWx5LXBsYW5ldC9Xb3JrYXJlYTo6Q2F0YWxvZzo6Q2F0ZWdvcnkvNTllZTQ5YjlmOTJlYTE0MTg2MDU4NTll.

Here is some reading
https://www.usip.org/publications/2022/03/armenia-azerbaijan-and-georgias-balancing-act-over-russias-war-ukraine
https://english.alarabiya.net/News/world/2022/03/28/Armenia-Azerbaijan-tensions-surge-with-Russia-distracted-by-Ukraine-war

I was there for a week in 2019 and recommend it highly. We traveled around the country and overnight in the disputed territories with a guide and driver, a private group of 9 friends and family.

We might go some day, Armenia was the first Christian country in the World.
Also, would be a good time to take in Georgia?

Georgia, yes. We spent a week there, too. The drivers, vans, and guides traded off at the border as we walked across. Everything was arranged by an agency in Yerevan, Armenia.
In both countries, the geography is spectacular. The religious sites are stricter about women's dress in Georgia. There were tourists from neighboring countries, Russia, Iraq, but also from as far away as New Zealand. There is more outside influence in Armenia since so many Armenians live outside the country. We saw new settlements of Armenians displaced from Syria, unfortunately in the disputed territories, which recently had its own war. Georgia was different in many ways.

Plane tickets to Armenia sell out quickly the moment they go on sale, so you have to be fast.
We were already in France when we flew Paris to Tiblisi with our group. The return was Yerevan to Paris and on to Indianapolis, a long day.

Despite the links above Armenia, Georgia and Azerbaijan are tops on my list. If not this year, then next year, at least one or two of them. Georgia I would feel a bit better about if done during the war in Ukraine as that keeps Putin busy. After that I am afraid Putin will want to expand his occupied territories in Georgia.

Anyway, Baku and Kutaisi are both relatively quick and cheap flights out of Budapest and plenty of seats in June. Flights from Istanbul on Turkish Air to Tiblisi are plentiful and available in June (the month I checked ... my next trip back here). Maybe fly US to Tibilisi on Turkish Air, then Kutaisi on to Budapest on WizzAir. With an open jaw Turkish Air ticket (IAH to KUT and BUD to IAH) I suspect that's maybe $1000 to $1250.

Just recently looked into tickets to Armenia - plenty of options available, including low-costers. No restrictions, no visas... Good beer (and even better cognac - well, I guess it goes by "brandy" now). Seriously considering getting those tickets to get a glimpse of Armenian summer.

Just to complete the loop - i did get those tickets, and Yerevan is great (as is the rest of the country) and has changed a lot since I was last there (doesn't quite apply to the other places I've visited).
May have to disagree with the "European" vibe - didn't pick up much of that - but for me that what made it special. Will return in a heartbeat.

PerilsofP, thanks for the update! Glad you had a good trip.