it could beat sputnik, but it happened not thanks to Eisenhower!
so far the story is tell, Von Braun had during test launch of his Rockets, two state officials in his back.
Who make sure that Von Braum not accidentally launch a satellite into orbit.
Next to that was the Battle "how launch the first US satellite ?" between US Army (von Braun) US Navy, NACA (need a rocket) and USAF (not interest to launch one)
In end the US Navy won, with untested Vanguard rocket and made Kaputnik on live US Television...
by the Way
Vanguard 1 satellite mass was 1.47 kg ! it was nicknamed "Grapefruit"
on the either hand Sputnik 3 mass was 1327 kg
Ike had his political reasons to support Vanguard--mostly related to recon sats. The US was developing plans for those since 1954, and the CIA in particular was eager to get them flying.
The issue was the question of the legality of flying satellites over Soviet territory. They shot U-2s down for airspace violation--what's to say a recon sat would soon find itself flying through a cloud of ball bearings if they tried a similar stunt with that?
The US needed a way to set legal precedent for satellite overflights of foreign territory--if the Soviets didn't shoot down an American satellite when it first goes up, then the US can claim an internationally recognized legal "Freedom of Space" similar to Freedom of the Seas. But a satellite operated by the US Army and designed by the chief missile designer for the US Army? That's got 'military' written all over it, and the Soviets would have been, in theory, well in their rights to shoot it down without international outcry.
Shooting down a 'civilian' science satellite like Vanguard is much harder to justify.
So, Vanguard got favor. When it went overbudget, the CIA even funneled millions of dollars into it. If the US could set a legal precedent for overflights of the USSR, then the CIA would be rolling in classified images.
The Soviets and Sputnik went and made that a moot point (overflying the US without getting shot down), so Explorer got the go-ahead soon enough after.
So, how does the Eisenhower Administration (presumably retaining a lot of key officials in the relevant positions, like Richard Bissell of the CIA) follow through in the 1960s?
It's clear that Ike's main goal was to put up reconnaissance satellites--all else was secondary. Here, the X-20 program has a certain place--before electronic transmission of images matured in the 1970s, the only way to get immediate images from the Soviets would have been to fly a Dyna-Soar and return the film after a single orbit. This was, according to
The Space Shuttle Decision, actually still considered a desirable mission for the Shuttle during the Nixon Administration, and led to that vehicle getting the large delta wings and high cross-range it had IOTL. Dyna-Soar would then stand a good chance of getting administration support in the 1960s, perhaps with some sort of MOL-like accessory program (a station with an even bigger telescope, which uses Dyna-Soars to return film canisters).
The Soviets would have their own counterpart programs (something like Almaz) as the decade goes on.
But how does this go on in the presumably Democratic administration of 1965-1969 (or 1973)? I suspect that the social upheavals of the latter part of the 1960s, and the economic stagnation of that period, would soon rear their ugly heads--nothing like OTL Apollo could get through a congress that might have even more Mondales and Proxmires than it suffered IOTL. And given that the technology for electronic transmission of reconnaissance images would mature by that time, the military reason for Dyna-Soar and *MOL would evaporate. I also suspect that their job-creation capacity would be lower than Apollo's was (given Eisenhower's rather cheapskate nature--thermonuclear weapons were emphasized for, among other reasons, their low cost during his administration). But before the unpleasantness of the end of the decade happens, a Democratic president, almost certainly running on an image of youth and vigor compared to the ancient Eisenhower, might see an opportunity to associate himself with high-technology wonders. Perhaps a reusable space launcher, to reduce the cost of launching the CIA's satellites...
But overall, the US manned space program will be much lower profile, and so have much less political clout (not that it has much IOTL). In the 1970s, with economic stagnation, and without the political base built up during Apollo, it will face much opposition from Congress.
That's how I see it, anyway.