unlike with big drums or disks, you don't need to carefully align the heads on a huge surface but can just let a small part of the tape come to the fixed head.
Sounds a bit like you never had to do so ;) I started my professional life in tape/disk service And believe me, there's a lot to be adjusted at a 34xx compatible tape drive head - not to mention what's needed to adjust the capstan drive and the vacuum chambers :)) (*1)
Spooled-up tape has the obvious problem of access time, which has always pretty much confined it to backups only,
Not really. that's maybe basic use of today – and what modern tape systems are optimized for. But there was a time when tape drives were a major component of external storage. In many cases the only one. If at all, disks where used for OS and software and maybe paging. All data was done with tapes (*2).
The heydays of tapes as main data media lasted through the whole 1970s with the IBM 3420 as cornerstone.
A 3420-8 (the top model in 1973) could read, write and seek data at 200 inch per second at 6250 bpi - which, at 9 track was also a byte. This gave a raw speed of up to 1,25 MiB/second. A maximum sized spool of 2400ft could hold 100..130 MiB (depending on block size) and be read/written/seeked in about two minutes. Pure rewind was even shorter at ~45 seconds. Further, blocks could be read in either direction.
These tapes were not meant as backup, but work media. Usually not the tape, but the mainframes CPU and application speed was limiting throughput and when used for disk backup, the tape had to wait for disk seek.
These tapes were blockwise rewritable (when keeping the same block size). From an applications view the differences to disk drives where rather minor - when keeping the tape nature in mind.
however that's not the only way to operate a tape – it's also possible to close the tape to a loop, so each part periodically passes the head. This isn't only theoretical, it has been the standard way to achieve echo in audio production.
And as well for cheap microcomputer storage with the stringy floppy and its most well known incarnation, the ZX Microdrive. A bit like the floppy story, as mainframe drives were far ahead.
Except they would presumably have been cheaper, and the loops could be made much longer than a drum circumference, to trade in latency for capacity – at arbitrary ratio!
What am I missing?
Beside that the fact that it has been done, that there is no speed gain.
If the tape is longer than a few meters, it needs to be spooled which in all cases will make it only single direction, resulting in exactly the same worst case access times as a bidirectional read standard tape, while being less effective in average seek time. Unless speed is greatly reduced, the mechanics would have needed the same effort as 3420 drives spend, thus being as expensive, without any improvement. With more simple mechanics, speed had to be greatly reduced to keep wear in check. Existing (cheap) stringy floppies did offer only 1/100 of the read sped of a 3420-8
Bottom line: What you wanted to do with loops was already achieved with standard tapes. They were a great storage (when used right), but did need exceptional effort to do so - and modern backup ones aren't in line with these requirements.
*1 - The Wiki got a rather good explanation of the workings in their 9-Track-Tape article - even though it's tied to a medium speed drive with vacuum chambers, but a conventional capstan.
*2 - I remember several customers who even in 1980 refused to switch for disk storage. The mainframe could be booted from tape and as data storage tapes could work at similar speeds as disks. Given a useful data organization ofc.