fast forward to 2017, and it isn't that fast anymore.
The 2007 Extreme (likely your model) should be capable of 150Mbps on 5GHz 802.11n signal. If you are connecting using 2.4GHz, it may be suffering from interference from neighboring networks. I recall it could use either 2.4 or 5GHz, but not both (simultaneous) as most newer routers can do. Unfortunately, there are still a lot of things sold that don't support 5GHz Wi-Fi (printers, IP Cameras, etc), so if you have these devices, setting the Extreme to 5GHz would disable wi-fi for some of these devices.
In the iOS version of AP Utility, you can enable a wi-fi scanner (Settings > Airport Utility > Wi-Fi Scanner). Then, open the app, click the Scan link at the top right and run a scan. It will tell you what channels and the RSSI (signal strength) for each network access point in range. The RSSI will be stated in a negative number, the higher the negative number, the weaker the signal (RSSI of -90 dBm is weaker than -50).
If your network SSID is using the same channel range as neighbors, you are contending for access to your radio signals. If you set your range different than the default, you may experience less contention and faster speeds. Also, most 2.4 networks gravitate to channel 6, right where microwave ovens, radars, wireless home phones, wireless keyboards, etc broadcast. Each range will actually use up to +- 2 channels, so setting channel 6 means it would use 4-8. The less overlap, and less contention from channel 6 will probably help. Given everyone has Wi-Fi these days, and newer routers often have stronger signals, the fact that it has slowed makes some sense, but it is probably not due to the router itself, but contention.
I name my 5GHz SSID slightly different than 2.4 (NETWORK vs NETWORK5G for example). This way, devices that can use 5GHz can specify that SSID and "forget" the 2.4GHz network and never try connecting to it. The fewer devices connected will also help with speeds.
5GHz is a shorter range and uses more channels, so contention is less likely but coverage distance can suffer.
If your ISP service is faster than 150 Mbps, then a newer router may help you take advantage of it. 802.11ac routers can generally offer 1Gbps or more on 5GHz signals and use tech to increase the number of simultaneous channels used to rx\tx data. The newest Extreme is capable of up to 1.3Gbps with ac capable devices (most 2012 or later devices support ac). But if your ISP service is 50 Mbps, you will never see internet speeds greater than 50Mbps, even with the fastest routers. If you do a lot of large transfers between devices on your network, newer routers might speed that up even if the ISP service is slower.
The other routers mentioned are probably very good devices as well. I favor the Airport line as my home is primarily Apple devices and I like the simplicity of it. I also use my Time Capsule for Mac backups. The newest Extreme can be used as a Time Machine backup destination with a USB drive connected to it.
Finally, count on obsolescence. 5G technology is expected to offer fixed Wi-Fi home internet service at speeds of 1-4Gbps at lower prices than fixed cable services offered today. So, most of us will be looking for internal Wi-Fi routers capable of extending this speed to all of our devices. The 802.1ac will be replaced soon with something that should be able to match 5G speeds. So, knowing that your investment is likely only for a 3-5 year solution, keep price in mind.
I find Apple's Refurb products are a great way to get their stuff at good prices. Right now, the current Extreme is available for $129 ($70 off) and offers the same 1 year warranty as a new device.
