Does White have any feasible way to break through this position? I recently played as Black in a 1+0 blitz game on Lichess in which the below position was reached.

[FEN ""]

1. e4 Nf6 2. Nc3 e6 3. d4 d5 4. e5g Nfd7 5. f4 Nb6 6. Nf3 Nc6 7. Bb5 Bd7 8. O-O g6 9. Bxc6 Bxc6 10. Ne2 Bg7 11. g4 Qe7 12. g5 a6 13. b3 Nd7 14. c4 b5 15. c5 Bb7 16. a4 c6 17. axb5 axb5 18. Rxa8+ Bxa8 19. b4 O-O 20. Be3 Qd8 21. Qc2 Qb8 22. Ra1 Bb7 23. h4 Qc7 24. Ng3 Ra8 25. Rxa8+ Bxa8 26. h5 Kf8 27. h6 Bh8 28. Kg2 Qa7 29. Qd2 Ke7 30. Kh3 Kd8


A lockdown position eventually ensued, and I offered a draw to my opponent, even though I theoretically could have won on time but I am a fair player, and they accepted.

The final position is one in which I see no foreseeable advance for either side. Black shouldn’t even think of making one, with a useless dark-squared bishop and an imprisoned light-square one. White has a knight pair, which could jump past Black lines, though it would be hard to do so. A good knight outpost is on a5. Their dark-square bishop is also useless. Therefore, White has an small advantage, but it would hard to due with dancing queens. If exchanged, then there is much less of a chance of an advance. 

If the game was continued between two other people who were given infinite time, could White, in theory, ever commence a winning breakthrough into Black’s camp, or can Black hold a draw? Ignore the 50-move rule if you must.
I don't see any way for White to make progress. There are a few ways but all don't lead to anything:

1) Maneuvering a knight to f6. This just gives Black an opportunity to exchange his bad h8-bishop. After ...Bxf6 gxf6, Black puts his knight on f8 and plays ...Ke8, defending the f7- and h7- pawns.

2) Playing the f5 pawn sac. The purpose of this would be to plant a knight on f4. But here Black just plays ...exf5 and again employs ...Nf8, defending the e6- and g6- squares.

3) Putting a knight on a5. This doesn't really due much and hardly warrants a response from Black.

I also let Stockfish think to depth 68. The evaluation it gave is 1.97, which sounds bad, except it gave this same evaluation before it reached depth 30. This suggests White has no way to make any progress, meaning most likely the result is a draw (though it doesn't prove it).

Btw, in a 1+0 game trying to flag your opponent in a dead drawn position is completely fine. Aiming to win on time in bullet is a huge part of that time control. I'd only considering playing on there an unfair practice maybe in longer time controls (say over 5+0).