Soviet spending was flat during the 1980s

Give Reagan his due, but the claim that he outspent the Russians (or that SDI caused an economic race the Soviets could not match) are overblown:

The Soviet Union's defense spending did not rise or fall in response to American military expenditures. Revised estimates by the Central Intelligence Agency indicate that Soviet expenditures on defense remained more or less constant throughout the 1980s. Neither the military buildup under Jimmy Carter and Reagan nor SDI had any real impact on gross spending levels in the USSR. At most SDI shifted the marginal allocation of defense rubles as some funds were allotted for developing countermeasures to ballistic defense.

If American defense spending had bankrupted the Soviet economy, forcing an end to the Cold War, Soviet defense spending should have declined as East-West relations improved. CIA estimates show that it remained relatively constant as a proportion of the Soviet gross national product during the 1980s, including Gorbachev's first four years in office. Soviet defense spending was not reduced until 1989 and did not decline nearly as rapidly as the overall economy.

To be sure, defense spending was an extraordinary burden on the Soviet economy. As early as the 1970s some officials warned Leonid Brezhnev that the economy would stagnate if the military continued to consume such a disproportionate share of resources. The General Secretary ignored their warnings, in large part because his authority depended on the support of a coalition in which defense and heavy industry were well represented. Brezhnev was also extraordinarily loyal to the Soviet military and fiercely proud of its performance. Soviet defense spending under Brezhnev and Gorbachev was primarily a response to internal political imperatives--to pressures from the Soviet version of the military-industrial complex. The Cold War and the high levels of American defense spending provided at most an opportunity for leaders of the Soviet military-industrial complex to justify their claims to preferential treatment. Even though the Cold War has ended and the United States is no longer considered a threat by the current Russian leadership, Russian defense spending now consumes roughly as great a percentage of GNP as it did in the Brezhnev years.

it's possible to argue a factual point that Reagan helped win the Cold War without invoking mythos. The real manner in which Reagan was critical was in trusting Gorbachev as a partner to peace - the Soviets were on the brink of collapse, and had Reagan been a cowboy the hard-liners might have opted to go out with a bang.