Fuckyouiway

A better router may help you with range and internal network speed but your wifi right now probably isn't the bottleneck... it's the connection.
 
To see if you might benefit from a faster router connect your modem directly to your best computer.  Then run TestMy.net.  Connect it back up to wifi and test the same computer again.  You'll probably need to unplug the power from modem each time when you switch back and fourth, they usually need to boot up and get the MAC address of the device it's connected to.
 
If you do that test and find your speed is much faster wired... then you might want to think about getting a newer router.  If the speeds are about the same... you can probably hold off unless you're having other issues with wifi.
 
I personally use Netgear.  A used Nighthawk R7000 for ~$70 would blow away whatever they gave you.  I also would never use a router or modem from my ISP and prefer to have them as two separate units.  Better for future upgradability, cheaper if something fails too.  I need to upgrade routers far more often than modems so having them separate is better.  Also better for interference. 
 
When I've tested even the supposed best ISP routers... they always under perform vs retail counterparts.
 
When it comes to wifi, testing right on top of the router isn't optimal.  You want to be a little away from it.  Maybe like 10 ft.
 
Here's an article about wireless best practices -- may help you get the most out of what you're working with. - wireless feng shui

If you're 2.5 miles from the "home office" (typically an MDF/IDF - main / intermediary data facility), you are outside the "sweet-zone" of ADSL.
 
Typically, (A)DSL sees maximum throughput at 2 miles or less from a MDF/IDF. Basically, the distance between your demarcation point (modem) and your ISP's closet.
 
You're only half-a-mile outside the sweet-zone, so I would still expect you to see reasonable speeds. That said, you might never see a stable 6Mbps, due to ADSL typically using older, buried phone / Cat3 lines. (Unless CenturyLink ran shielded coax/twinax copper or fiber, which I highly, highly doubt.)
 
The burst you see with TMN is likely because the lines are cool, so there's no electromagnetic field around them. Because ADSL is typically Cat3, it's often minimally (if at all) twisted, unshielded, and poor-cabling, so it's easy for the cabling to build an electromagnetic field that interferes with signal transmission. This field won't exist at an idle, it will only exist when data is being transmitted. Once the field has built up (often as quick as 50-2500ns) it starts creating cross-talk and signaling interference, slowing the throughput you can achieve. (Basically, error-rates go up, and as a result more of your bandwidth is spent on handling those errors.)
 
Unfortunately, without replacing the cabling, your speeds probably can't be improved much.
 
All that said, does CenturyLink offer 5Mbps? If so, I would downgrade to that and run some more testing. If their signaling is wonky, you'll see a similar drop (I would expect either 1Mbps or 833kbps drop when you switch if they have a signaling calculation incorrect). If you don't see a drop, save yourself the money and keep the 5Mbps connection. If you see a drop, document it (now, the 5 vs. 6, and then, the x vs. 5). If you still have a drop, call CenturyLink and explain the situation. I have no idea how well they'll assist you, but they should be able to make other accommodations (provide 6Mbps but bill at 5Mbps, for example).

Throttling at the "signalling level" as you suggest seems to be a pretty good guess going by the graph that I attached.  This is typical of what I get. I just wish they would throttle at what I'm paying for.  It's like buying something that costs $9.50 total, giving them $10.00 and not getting my change back.
 
I have only recently discovered TestMy.net. It's a fantastic service that's being provided and has explained my irritation with Centurylink over the years. It is only recently that the service has become stable where I live (west central Missouri) so I guess they're doing something to improve with the exception of providing faster speeds.  Every Mbps counts in a rural setting.

I live in Zurich,Switzerland, $35 for up to 1.25 MB/s.... in reality I pay $35 for 280 KB/s. That's Iway Ag for you folks

Lets see..... 125kbps down 101kbps up.
https://testmy.net/db/imGCMIxlD.87dTs5p2y

Note that the test result included this caveat:  633% faster than my average  US 818 kbps  (!!!!)  I live in northern New Mexico, 20 miles north of Santa Fe, and CenturyLink charges me $45 a month for 7 Mbps, which I never get.