Mode-locking in Cr:ZnS/Se lasers typically rely on Kerr-lensing (KLM) or a semiconductor saturable absorber mirror (SESAM). The former allows generation of shorter pulses, but, unlike the latter, does not support self-starting mode-locking. Here, we combine the advantages of these two techniques and demonstrate the SESAM-assisted KLM Cr:ZnS laser. Our self-starting oscillator generates up to 1 W of average power with 54 fs pulses at a central wavelength of 2360 nm. We identify a general limitation for further pulse shortening in SESAM mode-locked Cr:ZnS/Se lasers, which is related to the finite operation bandwidth of the semiconductor absorbers. In our experiment, we fully exploit the potential of commercially available GaSb SESAMs and fill their entire reflection bands. Furthermore, we compare the performance of a SESAM-assisted KLM laser with a pure KLM oscillator producing broadband, yet not self-starting, 33 fs pulses with 780 mW power. We also show that the choice of saturable absorbers has a negligible impact on the laser intensity noise, which is exceptionally low with sub-0.005% integrated noise.