There are several possibilities. "gatekeep" touched on a few. Many new radios have a foldback capability that reduces the output power, if the SWR is above a certain ratio (2:1 or 3:1). But the foldback should be very quick and not allow transmitting at full power for a couple of seconds. It's more likely something other than high SWR is the problem. While the Power Module could certainly be the problem, there are other possibilities.
I looked at the manual for the TK880H and it doesn't mention having foldback capabilities. Reading the circuit description, and looking at the schematic, I see it does sample the output power for stabilization and does have a thermal sensor. Its possible that the thermal sensor has failed and is shutting things down or, the Automatic Power Control (APC) circuit is getting incorrect information from the power sensors, and shutting things down.
IMHO, your best bet is to put it in a box and send it back to Kenwood for repair. If it was me, I would open it up and poke around a bit. But I don't know what kind of abilities you have.
Who ever told you that needs to educate himself on SWR and what it means. Here is a link to a very good group of articles on SWR and Reflections.
http://www.k6mhe.com/files/Reflect_rev1.pdf
A mismatch between the antenna and the feedline can cause some of your power to be reflected back towards the radio. It travels back down the feedline until it reaches the antenna connector on the radio. But it doesn't flow back into the transmitter and cause excess dissipation and other damage. The reflected power is then reflected back up the coax and to the antenna. The antenna will radiate some of that reflected power and the mismatch, at the antenna, will then reflect some of it back down the coax.
In the end all of the power that you send to the antenna will be radiated, except for the power lost in the coax. The reflected power is also subject to coax losses. There are on-line calculators available that can show you how much power is actually reaching the antenna, based on SWR, coax length, coax type, etc.
You originally said that you are using RG-58. That really isn't a good coax to use at the 440 MHz. RG-58 has about 12 db of loss at 400 MHz. A better selection might be some RG-6 (better) or RG-11 (best). Either of them will have less than 6 db of loss. You can often get these coaxes free from the satalite and cable installers. Yes, both are 75 Ohm cables, and may cause a little bit of a mismatch (SWR). But power loss due to mismatch is very small. Much smaller than the 6 db you gain from using a cable with lower loss. Try the on-line power loss calculators with RG-58 and RG-6/RG-11, and see the difference.
Of course, there is better coax, but I am just sugesting a, possibly, cheap upgrade.
Martin - K7MEM