Actually if you don't PvP, you can find yourself running out of quests as early as level 17. This game has a lot of grinding in it. It's ridiculous once you get in the mid to upper 30s. Especially since they made IV party only.
That simply isn't true.
There's 2 primary reasons why people run out of quests midway through a level.
The first is by Teaming up to do quests. The amount of XP earned is split and you will find yourself out of quests and needing to grind far earlier than if you had done them solo.
Every class is capable of doing all of their primary quests solo.. yes, even full support medics.
the other reason is that most people stack their quests and complete multiple missions with the same requirements at the same time. Example: 2 quests requiring you to kill Amphibyon Warrior being completed at the same time.
There's no reason to run them both at the same time. If you accept one, complete it, then accept the other one and complete it separately, you're increasing the amount of XP earned.
You aren't saving any time by grouping all your quests at once and completing them in one trip. All you're doing is increasing the amount of time you need to grind.
I've levelled many characters to the upper 20s without needing to grind at all by doing my quests one at a time.
I've levelled most of them to 21 without doing any PvP at all and still had leftover level 20 quests. I don't like doing the BGs without a mech or cyberskin, so I simply don't until after I get the skin quest in Mereholt.
To reiterate, there's no requirement to PvP when levelling and while the BG experience points will speed up your levelling, they are not required at all. You may, in fact, earn more experience running a dungeon in the same amount of time.