In which case I'm baffled. If that diagram is accurate then there's a contiguous VLAN from the router to each switch. I can't spot an obvious reason why the router would be unable to ping each switch, especially if PC 1 can.
As for the graphics, I cheated... 
https://www.draw.io/
I, too, find this behavior unexplicable and am inclined to blame it on some obscure feature of the cheaper switches. Apparently it is impossible to add tags to threads, so I edited the subject line to include the word "Netgear". Maybe someone else will stumble on it and give us a satisfying explanation.
At the very least, however, thanks to this discussion I learned an alternative way to manage the gs116e (the web interface), and a simple way to cook up a quick, nice-looking net diagram! Thank you! 
I took delivery of a new toy today: a TP-Link TL-SG105E.
It supports 802.1Q VLANs, up to 4094 of them.
It supports both DHCP and static IP addresses for the management interface (static 192.168.0.1/24 by default). Haven't yet worked out if the management interface VLAN can be changed (haven't read the instructions yet).
It has both web-based management (seems to be port 80 only, judging by an nmap scan) and a GUI management tool (not looked at it yet). Haven't yet found any evidence of a CLI.
It has an 8-port variant, which is half the price of the Netgear GS108Tv2.
Initial impressions are encouraging. If you're not worried about HTTPS management, these things look like reasonable alternatives to Netgear.
I'll see how it performs long-term; does it remain encouraging?
Quite. But I see its price is nearly identical to the gs108e's, so I guess it's more or less equivalent to a gs108e-v3 - or possibly just a little better, considering TP-Link's prices are a bit lower on average, and factoring in a few years of technological progress.
For what anecdotical info is worth, my latest TP-Link experience was with a switch (a 16-port model), and it wasn't great. There was a problem with VLANs. I don't remember exactly what it was - something along the lines of "can't be managed (or maybe even can't talk DHCP?) on some VLANs (tagged? Non-default? Can't remember)".
With my particular setup, I really couldn't work my way around this issue, so I had to return it and got the Netgear gs116e, which was a bit more expensive. By the way, the shortcoming was not documented.