Hi there and welcome.
The pick sound is part of what makes a guitar a guitar, and is not something most guitarists would want to get rid of. Not completely at least.
That said, there are two (possibly three) approaches I'm aware of that can pull it back into the sound a bit:
1. Use a bow. No, really: https://ebow.com/
2. Use lots of gain. Also varyingly called distortion, overdrive, fuzz, etc., but as much as you can possibly muster. My guess is that this is more likely to be the kind of sound you're after, which in turn suggests a metal sound of some sort. The idea is that the signal chain is being driven so hard that there's no gain left for the pick strokes to stand out.
(3. There is a third way, but I'm not sure you can still buy the kit you'd need, or indeed would want to even if you could. A noise gate with a variable attack time that can be set to kill the initial pick sound. TBH I'm not sure you can even buy one these days as they're a bit of a throwback to the days of noisy amps. Modern equipment suppresses the hiss better to start with so nobody really uses them anymore afaik - certainly not in stomp-box form. In any case it wouldn't work well with a continuous sequence of notes as it needs gaps to reset itself. Not really an option in your situation, and I only mention it for completeness as something you might want to look at further down the road.)
It's also worth saying that the Telecaster's characteristic sound is actually quite bright and pingy to start with (it's very popular with, for example, country & western guitarists for that very reason), so you're a little bit up against it. Doesn't mean you shouldn't try of course...