I found 2 processor Pentium III 1000 MHz (see photo). One of them is a later release and has a heat-distributing cover. However, they are both with the Coppermine core, so I do not need to modify the socket and change the Bios to switch i440BX to Tualatin.
Both processors excellently bother at a bus frequency of 140 MHz, but not one of them could not work reliably at 150 MHz.
For cooling, I bought a cooler intended for Socket 462 (Socket A). When it was delivered, it was twice the size of what was in my Socket 370, but there was not the slightest obstacle to its easy installation in the motherboard.
It has a bifurcated bracket attached to the 4 protrusions of the socket, in contrast to the radiator for socket 370, which is attached to only 2 ledges. So it seems that this is kept much more stable.
The image from the hard disk was successfully transferred to the IDE SSD and the partition was increased to 16 GB using the alignment operation. This inexpensive drive showed sequential read speed of about 85MB/s, which is 1 order of magnitude greater than the old disc, but the most bottleneck here is Hard Disk Controller - PIO Mode 5/DMA Mode 3 (33.3MB/S).
Probably a better solution is to use a separate expansion card in a PCI slot with a sata controller. But then the loading of an individual BIOS for such a controller will be added during the start of the computer, which can again worsen the start time of the device.