Jelly, 7 bytes

ẆSÐḟṪọ“

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To exclude the closing quote, this is a full program, not a function submission. The footer is just to make it work for the test suite.

+2 bytes to handle the empty edge case. this is definitely not the best way to handle the edge case but i'm too pissed off to deal with this shit right now (not related to the challenge)

Jelly, 7 bytes

ẆSÐḟṪȯ$

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Converted from a full program to a link for the same byte-count thanks to Jonathan Allan.

ẆSÐḟṪȯ$    Main Link
Ẇ          Get all sublists (unfortunately excludes the empty sublist)
  Ðḟ       Filter to remove elements with a non-falsy/non-zero
 S           sum
      $    Apply monadically to the resulting list of sublists:
    Ṫ      - take the last one (returns zero if the list is empty)
     ȯ     - logical OR; if 0 was returned, instead return the list
             of sublists, which is the empty list

Jelly, 5 bytes

ẆSÐḟṪ

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ẆSÐḟṪ    Main Link
Ẇ        All sublists, ordered by length and then position
  Ðḟ     Filter; remove elements with a non-falsy (non-zero)
 S       Sum
    Ṫ    Take the last sublist

Jelly, 7 bytes

ẆSÐḟṪọ“

Try It Online!

To exclude the closing quote, this is a full program, not a function submission. The footer is just to make it work for the test suite.

+2 bytes to handle the empty edge case. this is definitely not the best way to handle the edge case but i'm too pissed off to deal with this shit right now (not related to the challenge)

Jelly, 5 bytes

ẆSÐḟṪ

Try It Online!

Jelly, 5 bytes

ẆSÐḟṪ

Try It Online!

ẆSÐḟṪ    Main Link
Ẇ        All sublists, ordered by length and then position
  Ðḟ     Filter; remove elements with a non-falsy (non-zero)
 S       Sum
    Ṫ    Take the last sublist