It's possible. But then again, IOTL, CBS denied that the cancellation had anything to do with Asner's politics, citing a fall in ratings for the last two seasons as their primary rationale. We shall have to see when we get to future pop culture updates.
I agree with your assessment of the man.
ITTL, Rumsfeld has already had a long and successful career in Washington pre-1980. First selected to the US House of Representatives in 1962, he was then reelected three times before being nominated to finish Everett Dirksen's term in the Senate when Dirksen passed away in September 1969. Elected to a Senate term in his own right in 1974, Rumsfeld then managed to repeat the feat in 1980, despite hailing from a state (Illinois) that overwhelmingly favored Robert Kennedy for President. In addition to his new position as Minority Whip, Rumsfeld serves as the ranking Republican on the Senate's Armed Services Committee. This position has helped establish him as, arguably, the leading GOP expert on military issues. This too at a time when national defense is becoming an increasingly important campaign issue...
Though he might not identify with the term personally, Senator Rumsfeld's political ideology can probably best be described as
neoconservatism. Per Wikipedia:
"Neoconservatives typically advocate the unilateral promotion of democracy and interventionism in international affairs, grounded in a militaristic philosophy of 'peace through strength.' They are known for espousing disdain for communism and political radicalism."
To call Rumsfeld a "hawk" would be a dramatic understatement. ITTL, Rumsfeld was one of the leading voices against President Udall's proposed Strategic Arms Limitation Treaty (SALT) and helped to prevent it from being ratified by the Senate. He supports vastly increased military spending and using that beefed up military to enact America's national interest up to and including possible regime change, if necessary.
Domestically, like most Republicans, Rumsfeld generally supports a free market economy and capitalism that is less "restrained" than the post-war liberal consensus. That said, he favors "reducing" the welfare state, rather than eliminating it. Socially, Rumsfeld, a Christian and member of a Congregationalist church, is pro-life and supportive of "traditional family values". He believes strongly, for instance, that marriage is a sacred institution, exclusively meant to be between a man and a woman. He favors "law and order" policies toward crime and ramping up the war on drugs. In 1980, in order to get reelected, Rumsfeld was forced to moderate many of his positions, especially on economic issues. While certainly no liberal, Rumsfeld is unlikely to propose any serious cuts to popular New Deal/New Frontier programs in the foreseeable future.
All this, when taken together, places Senator Rumsfeld somewhere in the middle of TTL's big tent Republican Party. Generally accepting of government spending on "entitlements" like Medicare and Social Security (though he would, of course, like to make "reforms"), he is socially conservative and one of the leading hawks in the Senate. Rumsfeld is hoping to position himself as a Nixon/Reagan hybrid - possessing all the supposed foreign policy chops of the former, as well as the winning smile and charisma of the latter. If this goes according to plan, he might just become the next face of the Republican Party.