I am a semi-semi-pro in mtg.
It's certainly NOT pay-to-win at semi-casual competitive tournaments*. 80% of the contestants own, or can borrow, a competitive deck. There is certainly a steep cost to entry if you own nothing, but there is a lot you can do to circumvent that. Just google it if you're interested, as I won't go into extreme detail here.
Booster draft: Grab 7 mates, chip in $6 each to buy a booster box** and play arguably the most skill-intensive format available. Keeps you entertained for at least 4 hours and everyone goes home with on average $4 worth of cards (which helps towards a Standard Constructed deck, or mitigates the costs). Sure it's more expensive than a boardgame, but not by an obscene amount.
Sealed: Grab 3 or 5 mates, chip in $12 each to buy a booster box** and play Sealed. Keeps you entertained for at least 4 hours and everyone goes home with on average $8 worth of cards (which helps towards a Standard Constructed deck, or mitigates the costs). Sealed is often described as a luck-based format, but that is overstated. I will beat you 9 out of 10 times.
**Booster boxes are about $80 and contain 36 boosters. You need 24 for a draft, so you keep 12 for next time (and I adjusted the price accordingly).
Standard Constructed: I would advise against getting into this until you own a good couple of cards due to playing Limited (Draft/Sealed, as above). This format is secretly the most expensive, because decks only last for a couple of months before cards are no longer legal to play in Standard.. And their value plummets a month before this rotation. Unless you know what you're doing it costs around $200 per year to stay competitive here, so this would qualify as pay-to-win (though there are some competitive budget options, usually).
Modern Constructed: Yes, you do need a starting budget to enter. While there are competitive budget alternatives for around $50 (and can later/slowly be upgraded to a pro deck), pro decks start at $300-500. However, Modern decks tend to last for many years. I've played mine for 3 years now, for instance, and it still sells for roughly what I bought it for. If I sell now my deck would've been for free, if you look at it that way!
Casual Contructed: This is pay-to-win, unfortunately. Of course there are many viable budget options, but imho the best way to mititgate the problem is to find a good set of deck construction rules (such as Pauper).
*'Semi-casual competitive tournament' may sound daunting, but there are many 'GPT's and 'pPTQ's being organised every weekend. Only about 50% of the participants care about winning the thing and everyone else is just there to have fun! Also, the pairings are Swiss i.e. if you lose, next round you play someone who also lost (and has the same record). So the more you lose, the worse on average your opponents become :) It's fun.