From what I hear about globular clusters, they are primarily composed of very hot giant stars, which are not the most conducive for life as we know it. Main sequence stars like our own, due to their relatively long lifespans and generally weaker radiation, are a much better candidate.
So, is it possible for a globular cluster to exist, but one composed primarily of small main-sequence stars such G-type, M-type and K-type dwarfs?
The opposite is true. You should check your sources. Globular clusters do primarily consist of small main-sequence stars.
Globular clusters are believed to be formed in the halo of the Galaxy at the same time as the Galaxy formed itself (although their formation still is poorly understood). Regardless of their formation process, any massive star inside the globular clusters will long have gone supernova or red giant and ceased to exist (or turned into white dwarfs). So virtually every star you see in a globular cluster is a sub-dwarf main sequence star, G-type with a life time exceeding 10 billion years being the most massive which could still be around.
Most stars in a cluster form at the same time, hence the most massive ones, which have the shortest lifetimes, will have evolved onto the giant branches, while the majority of the stars sits happily on the main sequence.
The branch-off point from giants to main sequences is used to get a rough estimate of the age of the cluster, this method is called isochrone dating and used widely.
On a different note, the Arecibo message was sent to the cluster M13 in 1974 for exactly the same reason, to maximize the number of main sequence stars in the beam width of the sending Arecibo radar emitter.
So, unfortunately, there are some imprecisions in the answers to this question.
Indeed, it is true that globular clusters are mostly composed of main-sequence (dwarf) stars just like @planetmaker said. The reason for that being globular clusters are old, usually at the level of >10 Gyr. For the Milky Way, see the so-called age-metallicity relation of globular clusters below.
Data for the globular clusters is from Kruijssen+2019.
Since these clusters are so old, the most massive stars had enough time to evolve out of the main sequence. Hence, these are either on the giant phase or have already died leaving behind compact remnants (e.g., white dwarfs, which are easily detected in these systems). This is the reason why globular clusters are NOT mostly composed of "very hot giant stars", because these have already died. Indeed, this is what @AtmosphericPrisonEscape described in their answer.
All of this means that globular clusters, because all stars were born basically at the same and with the same chemical composition, the star-to-star differences in their HR diagrams are exclusively associated with variation in the mass of its stars. The distribution of stellar masses in a given stellar population depends on the initial mass function of the system, which would be a whole other discussion, but the bottomline is that globular clusters contain stars of various masses. Therefore, these occupy different locations in HR diagram, i.e., different evolutionary stages. See in the HR diagram below how the globular cluster contains stars of all kinds of evolutionary stages, from the main sequence and turnoff to giants and white dwarfs. For this figure, I specifically selected the closest globular cluster to us (M4), which allows us to see a very prominent main sequence.
HR diagram of M4 in Gaia bands based on member selection by Baumgardt & Vasiliev 2021.
All right, so this covers the original question. What concerns me is the assertion that globular clusters are "believed to be formed in the halo of the Galaxy", which is NOT true. Halo globular clusters are deposited into the Milky Way during the process of its hierarchical assembly. The hierarchical paradigm basically states that galaxies grow in size through a succession of merger events with other galaxies. In the case of a massive galaxy such as the Milky Way, it accreted several small dwarf galaxies throughout its history, which brought in their own globular clusters now scattered across the halo. See, e.g., Massari+2019.
Of course, there are several globular clusters born "in situ", in the Milky Way itself. These are mostly confined to the Galactic bulge, although we see some of these systems on disk orbits as well (see the above-linked reference). These in situ globular clusters are usually more metal-rich at the same age, which can be seem from the age-metallicity relation above (blue line). Those globular clusters accreted from dwarf galaxies ("ex situ") have lower metallicity at the same age (red line from Limberg+2022).
Hope this answer not only contributes to the original question, but also fixes the imprecisions in other's answers. Also, I guess its always good to add refereed references. Finally, I made all the figures myself, so all should feel free to use them.
