The memory channel info you posted looks functional for simple FRS-01 operations.
Icom has been improving the user-friendliness of the software the past few years. Right-click on any item and context sensitive help may be an option.
SQUELCH
Radio-wide Squelch is set in the Common | Set Mode screen. A channel with SQL Tight off (blank) uses the default radio-wide value. Depending on the radio version, squelch can be 0 to 255, or 0 to 9; 0 is wide open while 255 (or 9) is full tight. Older Icom software versions default squelch to 0 (wide open). I like to start just below halfway, but every radio and circumstance is different and as long as it works there is no wrong answer. 125 (or 4) usually is about 0.25uV or minus-122db, and that's pretty sensitive.
SQL Tight Offset is set in Common | Common (all the way at the bottom). Icom defaults this to the maximum value. SQL Tight ON for a channel uses simple math: the squelch setting on that particular channel is the sum of the radio-wide Squelch and the SQL Tight Offset values. Anything plus the default "all she's got" is pretty tight.
PROGRAMMING
Programming one of the "P" buttons for "User Set Mode" in Common | Key & Display will give you instant access to radio-wide squelch setting on the fly. If you don't have a "P" button programmed pressing "Up" and "Down" together when you turn on the radio temporarily makes P0 User Set Mode. Either way, after a LONG initial press each each subsequent short press scrolls down through the Enabled (not Inhibited) items listed on the software's Set Mode screen. UP/DOWN changes the values. Details are in the manual, but trial and error won't hurt anything. Do nothing for ten seconds, another LONG press or power cycle the radio to get back to normal.
Per RR, 462.5625 is both GMRS and FRS, and the 4-digit "5625" makes it narrow. FRS has a 0.5 watt limitation; GMRS (with a license) is 5.0 watts. The F60 is about 4 watts on H, 2.5-ish on L2 and 1 watt-ish on L1. All are too hot for FRS.
MORE LIKELY...
Bear with me...
In Common | Set Mode, enable Battery Voltage. When you turn the radio on the sensed battery voltage will display briefly. 8.3-plus is full, 7.5-ish is half and below 7 requires charging.
If the radio is turned on when it is in a BC-119 charger it can actually run dead while sitting in the charger, even with a good battery, green light lying to you all the while. Icom has a new BC-190 charger that fixes this.
BC-119's may die completely, but seldom go bad; wall wart transformers are another story. If in question, any 1000mW (or bigger) 14-16 VDC center-positive wall wart will work. BC-119's also charge just fine plugged into your car. Check your junk drawer for a cigarette lighter to charger cord- just make sure it has the center positive polarity!
BC-119 chargers are designed for radio-turned-off charging. When you drop the radio in the charger it will bring the battery to 8.4 volts then all but stop charging until the radio is removed/replaced and the cycle restarts. A weak battery may jump up to 8.4 volts and shut down the charger, but contain very little actual power.
Why did I go through all this? F60's are not battery sensitive. They are protective of batteries- even new, good L-Ion's don't like being deeply discharged. By design the radio firmware will switch to very low transmitter power any time the input voltage is below 7V, and will completely power off at 6.6-ish. This can also be caused by a battery that has a surface charge (7-something RECEIVE volts), but no capacity (6-something under transmit load).
This is common with older batteries and similar to the problem you described.
There are fancy (read as expensive) battery analyzers, but try this:
1. Turn the radio off and put it in the charger until the light goes green.
2. Turn the radio on, note the battery voltage, make sure you are on a High Power programmed channel, transmit continiously for 30~45 seconds and watch the display.
3. If a small "LOW" indicator pops up at the top of the display when transmitting, the battery is too weak to provide full power. The processor has shifted the transmitter into low power to keep an excessive voltage drop (below 6-something) from wacking the battery and/or radio. After transmitting, the LOW indicator will normally stay on then go out after a few moments when the bad battery has surface charged itself back to a normal setting. The battery gauge will show something similar. This is an intentional design - replace the BP227 battery.
4. If you didn't see LOW, let the battery run down a bit and try again. Starting with more than 7.5 volts, going LOW after less than 30~45 seconds of transmit indicates a bad battery.
Also try: Remove the battery. Use a q-tip and alcohol and clean the radio and battery contacts. The waterproof contact shield seals water out, but I've also seen them seal gunk in and drop battery voltage under transmitting load.
ONE LAST THING
If the LOW indicator doesn't come on (and you have tried new or known good batteries with clean contacts) there's one more possibility: F60's have a pretty robust little transmitter exciter- I have seen several F50/F60's with blown transmitter power amp finals (usually caused by transmitting into a bad or no antenna) that sent 0.05 watts to my service monitor. Enough to make it a block or so but not much further. Again as in your description.
Bad news: If it isn't a battery or battery contact problem send them to Icom to get fixed.
Good news: $125 each will get them flat-rate repaired at one of the three factory-authorized regional repair centers, aligned, spit-polished and brought back to like new; have them replace the side connector board while they're at it. I have tons of these in rough service, including farm manure pumpers who drop them into pits full of pig... stuff. The street price for new submersible severe service radios is about $500- do the math.