Not necessarily....
Without NASA (and I agree, the only way to get rid of that Albatross is to butterfly away Eisenhower, preferably with Taft, et. al.), the various military contractors would have all pushed for multiple approaches to spaceflight, manned and otherwise. There is a fairly high probability that (once again, WITHOUT Eisenhower, even though this predates NASA), there would have been no Sputnik moment (the Army and Von Braun get a successful launch first), and you probably have a successful manned launch in 59/60.
The big changes come afterwards though. Yes, the military wanted space for recon first, and yes...they definitely knew that they could get that without manned space, but having manned space would give them lots of flexibility that simply wouldn't be possible otherwise. With multiple approaches to manned flight (including things like Dynasoar, as well as some of the more conventional capsule based concepts), you get a real infrastructure in LEO, with things like orbital fuel depots, garages, etc., not to mention the vast amount of institutional knowlege (zero-g operations, repairs, etc.) that only comes from multiple flights and lots of experience. All of this means a more robust space flight technology, with multiple players, both private and government.
TxCoatl1970 makes the excellent point that the Outer Space Treaty banned weapons in space, but I wonder how much the US would have been willing to go along with this in the early 1960s (when the treaty was negotiated) if they perceived themselves to have an advantage in the Space Race. Certainly things like orbital bombs and FOBS would have been outlawed (both sides saw those as unacceptable problems, hell even Kissinger acknowleges that) but that is not the totality of what the OST was about, and without it, lots of good things might come to pass. Without the OST, the 'space as the common heritage of mankind' nonsense doesn't happen, and there is a much bigger chance of commercial activity. Certainly the option of property rights makes lunar exploration more attractive (though given the costs and problems associated with it, I am not sanguine about how far that goes until the 80s or 90s), with even some possible attention to near earth asteroids.
The whole notion of ABMs leading to preemptive strikes is simply silly, to be honest with you. No ABM system under development in the 1960s (hell, even those in development NOW) had any chance of doing more than simply making attacks against SOME targets more expensive. When you are using nukes, even a few 'leakers' are enough, and the logical approach to an ABM system is to build more offensive missiles, preferably with larger payloads for multiple warheads or penetration aids. This is precisely what the Soviets did in response to Sprint/Sentinel, along with the development of strategies to go after the various components of the system (PAR and Battle management radars, etc.) in OTL. The Soviets continued a focus on 'heavy' ICBMs late into the 80s for precisely this reason, and they would have likely continued to improve their SLBMs for more flexibility as well. Given even a marginally useful ABM system, why would first strike (which would have to be overwhelmingly successful to be practical) be a viable response? If the ABM system degraded the first strike sufficiently to permit effective response, then the point of that strike is lost, whereas if you instead focus on a viable second-strike option (survivable forces) then an ABM system would have to be far, far more effective to have any real impact on the strategic calculus.
Finally, SPS systems are fun to talk about, but they are simply not viable with any technology we are likely to see for some time. The costs are unbelievably high, they must be placed in orbits high enough to require stupendous amounts of propellant, and effective (cheaper) alternatives are readily available. There are lots of reasons to go into space (once again, TXCoatl1970 mentions several), but SPS is not one of them...