Making sense of shares buyback

I am trying to proof out the fact that share buybacks does not increase company's value or shareholders' value.

Scenario:

Lemon Corp.

Asset - $100

Liability - $0

Equity $100

Sell 10% at book value to the public. Split into 10 shares. Value at $1 per share.

Company earns $20 in year one. Public shareholder's claim to the income - $2. Earning Per Share - $0.2

Company balance sheet at the end of year 1:

Asset $120 (Cash $20 / LT asset $100)

Liability - $0

Equity - $120 (Capital $100 / R/E $20)

Company decided to buy back 5 shares from the public. The stock price jumped from $1 per share to $2 per share due to the excellent first year earning. The company is now trading above book value.

Bought back 5 shares at $2 a share = $10 cash outlay.

Company balance sheet after the buyback:

Asset $110 (Cash $10 / LT asset $100)

Liability $0 

Equity $110 ($100 Capital/ R/E$ $20 / Treasury Stock $10 (contra account))

Share outstanding: 5 (represent 5% claim in income/profit)

Year 2, the company earned $20 again.

Public shareholder own 5% of $20, which equal to $1. There is 5 shares outstanding. Which equal to $1/5. Earning per share - $0.2

Company balance sheet after year 2:

Asset $130 (Cash $30/ LT Asset $100)

Liability $0

Equity $130 ($100 Capital/ R/E$ $40 / Treasury Stock $10 (contra account))

It seems like from an accounting standpoint, the company and its public shareholders can only realize the benefit of the buyback if and when the company and sell its treasury stock above what they purchased it for. Where the treasury stock sit (equity or asset side) makes no different to EPS unless they decided to sell it back to the public at a higher price or mark to market at a higher price.

So instead of buying back shares, why not give the cash back to the shareholder to decide?