A very broad definition of 'rap' would be - rhythmical speaking, as opposed to singing a tune. The instrumentation is not really a part of the definition.
The Beastie Boys were originally a punk band. They could play their instruments. The idea of using samples of other people's records was really quite new in the mid 80's. Technology was nearly but not quite ready for it. There were bands who used samples - & the laws governing the use of samples of other people's records hadn't been firmly established at the time.
However, there was a movement at the time which was using real rock musicians with rap vocals.
I don't know whether it was the first, but a major breakthrough record in 1986 was Run DMC & Aerosmith's Walk This Way
This was a true collaboration between a [sorry guys] fading rock band from the 70's & a new up-and-coming band & musical style. This ground-breaking combo of real rock music - the band played & sang this fresh for the collaboration, over a drum machine & some currently popular scratching, they didn't sample the old record - hit the perfect combination of rock & rap. Even people who didn't like rap or dance music but preferred their rock to be 'old school' liked this track.
It also happened as MTV was happening - in a time when to get on this station you 'had to have a rock guitar solo'. There were many complaints that black music could not get any airplay. [This was strenuously denied, but listen to some of Michael Jackson's output from this time].
So, along come the Beastie Boys. An ex-punk, new rap band…
Play your own instruments - I'm sure even the drums are live on the first album - & you have a slightly more raw [I wouldn't call it cash in…] rock/rap for a new generation - building on what had gone before…
meh, seems to be restricted - have to click through to YouTube
It was also popular, you may notice from these videos, to preface any music video with up to a minute of "story"… eventually people got tired of that ;)
In the year between the Beastie's first & second album, things were moving rapidly. Sampling was becoming more popular, the brief foray into rock/rap was rapidly fading in popularity… so their second album is 'more sampled'. Interestingly [although I was one of these so never really noticed at the time] fewer people cared. They were busy listening to newer bands who had started with a sample base, not transitioned from 'real instruments'.
The 80's was a period of rapid transition. One we haven't really seen since.