I'm thinking completely theoretically, so if the world doesn't work like finance theory, fine. But I just can't see any way in which leverage, debt, cash, etc don't substantially alter the market cap of a company.
Not necessarily. If shareholders believe the company with the less cash has a better future, that is if future earnings/growth expectations are higher for the company with less cash, then said company may be granted a dramatically higher market cap.
Two companies with identical balance sheets, identical growth rates, and identical net income numbers, can have radically different market caps. It depends on how shareholders regard the companies, what they think their futures look like (and sometimes it even depends just on more ridiculous things like hype / herd insanity, as eg with the dotcom bubble, in which Cisco was given an extraordinary multiple off its earnings).
Walmart's balance sheet & profit picture were far superior to Amazon's at the time when Amazon became larger than Walmart in terms of market cap. Shareholders came to believe that Amazon had a far brighter future, in regards to growth etc etc. Amazon has been granted an unusual valuation for a very long time, on the basis of expectations of future growth.
Would you rather have a $1 in cash and a $9 share in the first company, or a $10 share in the second company where $1 is the cash per share? The second case is more risky so you wouldn't pay the full extra $1 for the second share.
Of course it could also be the case that the spread is larger than $1, if the first company is too short on cash and there is a liquidity risk depressing the valuation.
Not if they're otherwise identical companies, it isn't. There may be reasons to expect a correlation between risk and cash holdings, or something to that effect, but ultimately a share in a company is a share of ownership of that company's assets. If a company didn't become $10B more valuable when given $10B in cash, where exactly did that value disappear to? Value doesn't vanish into thin air simply because it's now owned by a company.