I think that having the president pick a vice-presidential candidate is probably, on balance, a good thing. The powers and duties of the vice-presidency have increased and it makes sense that the vice president be someone the president can trust and work with. That's likelier if the choice is made by the presidential candidate than by others.
I actually think we could do without the vice presidency. If the president needs a deputy or second-in-command then make a separate, appointed, executive officer. Call him or her the "Assistant President" or "First Secretary" or "Chancellor," and have them go through Senate confirmation. That way you get someone who is more likely to be a truly valuable deputy. By making the vice president part of the election process you simply introduce all kinds of electoral considerations that really shouldn't be relevant if you want someone to simply be an able deputy.
And if the president dies, resigns or is removed from office, the office can devolve to the First Secretary on an interim basis and then you could call a special election (so long as there's more than a year left in the term).
There are a couple old articles from The Atlantic that advocate abolishing the vice presidency and instituting a special election requirement. One is by Arthur M. Schlesinger, Jr., from 1974, and the other is from 1947. Schlesinger advocates a general election to be held within 3-6 months for the remainder of the presidential term, while Wilmerding (the author of the 1947 piece) argues for simply holding an election for a full 4-year term in concurrence with the next congressional election. So, for example, there would have been a presidential election in 1974.
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http://www.theatlantic.com/unbound/flashbks/pres/wilmer.htm - The 1947 article
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http://www.theatlantic.com/doc/197405/schlesinger-vice-presidency - Schlesinger's 1947 article.
