I guess it’s like going to the Vatican ONLY if you can do it without out others there taking pictures.
I have walked by and seen no one at the door and I have walked by and seen a very long line to get in.
About 10 years ago I went with my two college sons. They got bored with me sitting and people watching so they got up and returned with a chess board. We were there 3 more hours. Can’t remember which son won, but seeing them there playing chess just fit the ambiance. A good memory.
If you want decent coffee and maybe, if they aren’t too busy, a conversation with a local then the Lions Locker. Nothing architecturally special about it, but very “real”. The guy in the black shirt and the young lady behind the bar are two of the family that own it: https://www.lockerbudapest.com/gallery. Say good morning, ask how their day is and sit back and enjoy the conversation (if they aren’t real busy).
For more traditional turn of the century coffee houses (but these are mostly tourist attractions these days … generally for good reason. They are lovely.
DISTRICT V
Central Grand Café (4.8), https://maps.app.goo.gl/czFuAfay1dq5Vvin7, Established in 1887.
Café Gerbeaud (4.2), https://maps.app.goo.gl/tdtPPBbUCBWeQLvq5, Established 1885.
DISTRICT VI
Művész Coffee House ** (4.4), https://maps.app.goo.gl/qnr7RjzPDkERYUbP9, Gracing Andrássy út since 1884.
**Zsolnay Kávéház (4.7), https://maps.app.goo.gl/xAd8YKyy1tWRg9ac9, Located in the Radisson Blu Béke Hotel, among the oldest hotels in Budapest. Part modern and part Art Nouveau. Coffee, sweets, meals and drinks. The mosaic at the front door, added in 1947, is György Szondi, the heroic knight captain who fought the Ottomans in the 16th century.
Maybe more to your liking might be: Auguszt Cukrászda Belváros, https://maps.app.goo.gl/XpgWKo2chT66nyfc9, don’t forget to check out the courtyard shops.