For engine vs engine automatic matches, the most practical UI is probably still Arena
As long as the engine is UCI/Winboard compatible, the vast majority are, you can add it to the list of available ones, and then run a 1vs1 match or a multi-engine tournament
Arena chess GUI sounds like it would work for you. The UI is a bit clunky, but it's free and allows you to run whatever UCI (or Winboard) engine you want with the analysis arrows. It's what I use for more serious engine analysis because the lichess engine isn't that powerful and I don't want to pay for Chessbase.
I use a chess program called arena for offline analysis. It's probably not the best one in existence, but I like it because the engine analysis tells you stockfish's best move, making it closest to the lichess "best move" arrow I have seen in any analysis software. I have tried other chess software like scid, but I couldn't figure out how to get them to output the engine's best move in the current position. Here is the website for arena - > http://www.playwitharena.de/
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Although, I recommend you just put the pgn into the lichess analysis window. Honestly, the lichess local analysis function is the best analysis software I have come across so far. It uses stockfish 11+, has a clean interface, and allows for easy adding of variations. If you want to annotate and stuff, you can create a lichess study with the game pgn. The only bad thing with lichess is that you need internet to access the website, which is why I went looking for an offline alternative.
The best GUI for the stockfish chess engine that I have found is lichess.org.
However, if you need something offline, Arena Chess GUI is the best thing I have found. The UI looks a bit dated, but it is easy to configure and the analysis is easy to understand if you have used the lichess local analysis board.
Among other GUIs, Arena has a lot of options in regard to engine vs engine tournaments, including preparing a set of partial games/positions from where each game will start, have each engine play the same position with reversed color and so on.... and of course, it supports pretty much all common UCI/WB engines, so you can pair them to your liking
I don't think it supports asymmetrical time controls when playing automatic tournaments, however
Normally the chess GUI and engine are downloaded separately.
Here is stockfish 12: https://stockfishchess.org/
If you google chess gui there are several choices.
Here's the first link that google returns and it works fine for me:
Here's a review for that GUI and for others:
https://hobbylark.com/board-games/Top-5-Free-Chess-GUIs-for-UCI-Engines
No software I know of has that kind of tree visualization. But for windows both ChessX and Arena Chess are pretty good, and you can definitely view and edit PGNs.
I think we're still at cross purposes.
Thank you for the clarification that books are not bundled into Stockfish. I don't believe I ever said they were, and I don't think it is relevant, but if it clears things up for someone it is helpful.
In this instance, OP was using Stockfish through Arena. By default, Arena comes with an opening book (as explained at http://www.playwitharena.de/). By default, Arena will choose its opening moves from that opening book. That explains the behaviour that OP was seeing, and how to get the behaviour OP wanted.
I'm also not sure your explanation is quite accurate. Many inter-engine competitions use an opening book - certainly the TCEC does - and they do so precisely to get the behaviour OP was complaining about, i.e. a diversity of opening moves! TCEC themselves explain that they use opening books because otherwise many games would open in exactly the same way each time. The reason that engines do not need to come bundled with an opening book is that the UCI interface assigns that role to the GUI, and I suspect that has more to do with modular design and avoiding duplication than the needs of chess engine matches - but I haven't gone into the history to check.