Is it possible complete Hollow Knight without beating the False Knight?
After the False Knight's first rage attack you can break the wall/gate on the left of the arena to escape the fight without beating him. So the answer to the first question is yes.
But I feel like you're more interested in whether you can actually beat the game without Vengeful Spirit, and I don't think that's possible. Even the Low% All Glitches speedrun category, which beats the game in 8% still picks up the Vengeful Spirit (of course that doesn't mean it's required, it just means picking it up is required to beat the game in 8%).
There was a post on reddit a year ago about trying to beat the game without Vengeful Spirit and it wasn't possible at the time. Though there is one comment saying that there's a frame perfect trick to kill the Elder Baldur without it, so maybe it is just barely possible?
There is currently a viable strategy.
It can be summarized as:
- Farm geo and buy Lumafly lantern (Husk guard above Stagway station is a good source).
- Enter Crystal Mountain through the dark passage, follow right, towards Crystallized Mound.
- Drop into the crevice right before Crystallized Mound, entering Resting Grounds.
- Through the Dream Nail sequence, enter Seer's hut
- Obtain Dreamshield (two platforms below Seer's hut). Equip it.
- Return to Dirtmouth and travel to the entrance to Greenpath.
- Use the Dreamshield to kill the Elder Baldur:
- stand just barely near enough that it closes up.
- when the shield is nearly exactly to the left, step slightly away from the baldur so it opens up
- immediately bounce the shield into the baldur with your nail
- repeat 12 times.