The leader-less state (and associated blank tape at head and tail) is because these are dubs, not masters. Once the show was recorded, the distribution dubbing was accomplished outside the recording studio itself, since there were probably upwards of 100 or so recipients of the tapes (vs. LP, cassette?, or network distribution). Dubbing likely took place on a Garner or Magnefax-type duplicator: (Do a Google image search on magnefax for some other views.)
...which was commonly used by studios for 1/4" dubs in medium volume. It would spool tape off of pancakes, and directly onto the reels you have. Tails-out, for print-thru protection. Common capstan for speed accuracy. No one would stop and leader-up a dub, unless the station itself wanted to do it. But for a one-play program, they would just cue it up and roll it. The head and tail are so long because the machine runs at 60 ips (play and record on same capstan/same speed) so it's flyin'. And back then, tape was cheap. (Similar machines would be used for music tape duplication in that time period, too. Higher-end dub houses would use individual machines; sound quality could be higher, but a more complex project due to individual machine maintenance matters.)
The same type of Magnefax/Garner system was used for medium-run cassette duplication, from pancake to pancake (yes, often without flanges on either) that was subsequently loaded into C-0 cassette shells by automatic splicers. Out-of-band audio tones were used to determine when program began and ended.
Either system could be configured with either a reel-based master (just right off the mastering machine) or a loop bin (AKA bin loop) system where the master tape was in a box, without reel, and edited into an endless loop. It could get ugly, but it was very fast.
The likely arrangement for a weekly radio show was reel-based masters, probably mixed on-the-fly (no multitrack + mixdown) and mastered at 7.5ips, then leadered-up and straight to duplication. (They might have rolled an archival 15ips simultaneously.) The show format is basically Casey does intros and segues, then timings are taken and the music is rolled in. (Casey probably didn't sit thru the music himself, but did a bunch of V/O and it was put together without him.) If the engineer makes a mistake in timing or levels, just back up a bit, try again, and edit the master as required as you're done. These were the days of mostly non-automated studios, no timecode controllers for audio work, definitely no digital audio workstations or anything. Voice talent skills, engineering skills, and a good timing rundown for script length, tune intro/outro length, and overall show length ruled the day. Pop in the national spots and promos, leave holes for the local ads, (or note places to stop tape) then dub and ship. 3 or 4 hour show probably took two full shifts to dub and prep for shipping. Probably a half-day in recording V/O, then a day putting the show together.
Chip
On edit: Sorry, change all the "Casey" bits to whoever the host of that show is. Thinking American Top 40, etc...