He's not pushing the vax. He's taking credit for getting them finished earlier than anyone thought possible. You have to view things through the lens of a business person (or a man, in general) who viewed the pandemic as a problem to be solved.

Trump approached it from any angle that he saw as a potential solution, which is why he hyped hydroxychloroquine the minute it was determined to be a potential benefit. Same with ivermectin, the ultraviolet light treatment, and the same with the vaccines. The vaccines are of course the only one that the media would allow. It's the same reason he pushed for shutting down travel immediately. He went for any and every potential solution possible.

He views the speed at which they were created as an achievement that he can take credit for, nothing more. He's obviously not a doctor, so he doesn't know the details behind the vaccines.