Iptables and counters

Remove the three rules you have above and try adding this:

$IPTABLES -t NAT -A POSTROUTING -s 192.x.y.a -j SNAT 192.x.y.b
$IPTABLES -A FORWARD -s 192.x.y.a -J ACCEPT

And enter the following command:

echo 1 > /proc/sys/net/ipv4/ip_forward

You should also have a rule similar to the following in your firewall:

$IPTABLES -t FILTER -A INPUT -m state --state ESTABLISHED,RELATED -j ACCEPT

That will take care of all established/related connections, instead of making a rule for each one.

What am I doing wrong?

Probably nothing. It's the way packets traverse Netfilter.

Check this diagram as a reference:

(Source: Iptables Tutorial 1.2.1 by Oskar Andreasson, 2006)

I cannot find them in the counters of any other chain or table

SNAT is a final Netfilter target, packets will not show up in the same chain afterwards. The POSTROUTING-chain in the nat-table is the absolute final table a packet can traverse the Netfilter-framework. Tcpdump is attached to a fairly ealier stage, I think in mangle/POSTROUTING.

and cannot see them on any interface using tcpdump. That's because tcpdump sees the packet's before they are getting SNATed.

Is something actually going wrong? It sounds like perfectly normal Netfilter-tcpdump-oddities.

Edit: Your SNAT statement happens in the end. Maybe you need to insert it before the -o wanif -j SNAT --to-source $WAN_IP statement. Since I have no more details, I can't tell if it's a mistake or intentional.