Is it possible complete Hollow Knight without beating the False Knight?

The only thing that the False Knight blocks off is Vengeful Spirit and the City Crest. I know that the City Crest is not needed, since the City of Tears can be accessed from Deepnest via the tramway and the Ancient Basin.

I'm currently stuck in Crystal Peak, which I entered by grinding Geo to get the Lumafly Lantern, negating the need for Desolate Dive. If I can get the Crystal Heart, I think I can get to Greenpath via the Howling Cliffs (assuming none of the platforming there needs the Mothwing Cloak). My issue is, I think I need the Mantis Claw to get to the Crystal Heart.

1. Is it possible to get the Crystal Heart before defeating the False Knight?
2. If so, can the rest of the game be completed without the Vengeful Spirit?
3. Otherwise, is there another possible route?

Solution 1:

There is currently a viable strategy.

It can be summarized as:

1. Farm geo and buy Lumafly lantern (Husk guard above Stagway station is a good source).
2. Enter Crystal Mountain through the dark passage, follow right, towards Crystallized Mound.
3. Drop into the crevice right before Crystallized Mound, entering Resting Grounds.
4. Through the Dream Nail sequence, enter Seer's hut
5. Obtain Dreamshield (two platforms below Seer's hut). Equip it.
6. Return to Dirtmouth and travel to the entrance to Greenpath.
7. 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.

Solution 2:

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?